aa r X i v : . [ m a t h . A P ] J a n THE CALLIAS INDEX FORMULA REVISITED
FRITZ GESZTESY AND MARCUS WAURICK
Abstract.
We revisit the Callias index formula in connection with supersym-metric Dirac-type operators H of the form H = (cid:18) L ∗ L (cid:19) in odd space dimensions n , originally derived in 1978, and prove thatind( L ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X i ,...,i n =1 ε i ...i n (0.1) × ˆ Λ S n − tr C d ( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x i n d n − σ ( x ) , where U ( x ) := | Φ( x ) | − Φ( x ) = sgn(Φ( x )) , x ∈ R n . Here the closed operator L in L ( R n ) b n d is of the form L = Q + Φ , where Q := Q ⊗ I d = (cid:18) n X j =1 γ j,n ∂ j (cid:19) I d , with γ j,n , j ∈ { , . . . , n } , elements of the Euclidean Dirac algebra, such that n = 2 b n or n = 2 b n + 1. Here Φ is identified with I ⊗ Φ, satisfyingΦ ∈ C b (cid:0) R n ; C d × d (cid:1) , d ∈ N , Φ( x ) = Φ( x ) ∗ , x ∈ R n , there exists c > R > | Φ( x ) | > cI d , x ∈ R n \ B (0 , R ) , and there exists ε > / α ∈ N n , | α | <
3, there is κ > k ( ∂ α Φ)( x ) k κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | = 2 , x ∈ R n . These conditions on Φ render L a Fredholm operator, and to the best of ourknowledge they represent the most general conditions known to date for whichCallias’ index formula (0.1) has been derived.We also consider a generalization of the index formula (0.1) to certainclasses of non-Fredholm operators L for which (0.1) represents its (generalized)Witten index (based on a resolvent regularization scheme). Date : August 2, 2018.2010
Mathematics Subject Classification.
Primary 47A53, 47F05; Secondary 47B25.
Key words and phrases.
Fredholm index, Witten index, resolvent regularization, Dirac-typeoperators, Callias index formula.To appear in
Springer Lecture Notes in Mathematics . Contents
1. Introduction 32. Notational Conventions 103. Functional Analytic Preliminaries 134. On Schatten–von Neumann Classes and Trace Class Estimates 205. Pointwise Estimates for Integral Kernels 286. Dirac-Type Operators 427. Derivation of the Trace Formula – The Trace Class Result 498. Derivation of the Trace Formula – Diagonal Estimates 579. The Case n = 3 7410. The Index Theorem and Some Consequences 7811. Perturbation Theory for the Helmholtz Equation 8612. The proof of Theorem 10.2: The Smooth Case 9413. The proof of Theorem 10.2: The General Case 10914. A Particular Class of Non-Fredholm Operators L and Their Generalized Witten Index 114Appendix A. Construction of the Euclidean Dirac Algebra 121Appendix B. A Counterexample to [22, Lemma 5] 126References 132HE CALLIAS INDEX FORMULA REVISITED 3 Introduction
If pressed to describe the contents of this manuscript in a nutshell, one could saywe embarked on an attempt to settle the Callias index formula, first presented byCallias [22] in 1978, with the help of functional analytic methods. While we triedat first to follow the path originally envisaged by Callias, we soon had to deviatesharply from his strategy of proof as we intended to derive his index formula undermore general conditions on the potential Φ in the underlying closed operator L (see(1.4)), but also since several of the claims made in [22] can be disproved.Before describing the need to reconsider Callias’ original arguments, and beforeentering a brief discussion of new developments in the field since 1978, it may bebest to set the stage for the remarkable index formula that now carries his name.For a given spatial dimension n ∈ N , we denote the elements of the EuclideanDirac algebra (cf. Appendix A for precise details) by γ j,n , j ∈ { , . . . , n } . Onerecalls in this context that for n = 2 b n or n = 2 b n + 1 for some b n ∈ N , γ j,n satisfy γ ∗ j,n = γ j,n ∈ C b n × b n , γ j,n γ k,n + γ k,n γ j,n = 2 δ jk I b n , j, k ∈ { , . . . , n } . (1.1)With the elements γ j,n in place, one then introduces the constant coefficient, first-order differential operator Q in L ( R n ) b n by Q := n X j =1 γ j,n ∂ j , dom( Q ) = H ( R n ) b n , (1.2)with H m ( R n ), m ∈ N , the standard Sobolev spaces. One notes in passing that Q = ∆ I b n , dom( Q ) = H ( R n ) b n . (1.3)Next, let d ∈ N and assume that Φ : R n → C d × d is a d × d self-adjoint matrix withentries given by bounded measurable functions. We introduce the operator L in L ( R n ) b n d via L : ( H ( R n ) b n d ⊆ L ( R n ) b n d → L ( R n ) b n d ,ψ ⊗ φ (cid:16)P nj =1 γ j,n ∂ j ψ (cid:17) ⊗ φ + ( x ψ ( x ) ⊗ Φ( x ) φ ) . (1.4)Given (1.2), we shall abreviate Q := Q ⊗ I d = (cid:18) n X j =1 γ j,n ∂ j (cid:19) I d , (1.5)and, with a slight abuse of notation, employ the symbol Φ also in the context ofthe operation Φ : ψ ⊗ φ ( x ψ ( x ) ⊗ Φ( x ) φ ) , (1.6)(see our notational conventions to suppress tensor products whenever possible, col-lected in Section 2 and in Remark 2.1). Thus, we may write, L = Q + Φ . (1.7)The associated (self-adjoint ) supersymmetric Dirac-type operator H in L ( R n ) b n d ⊕ L ( R n ) b n d is then of the form H = (cid:18) L ∗ L (cid:19) . (1.8) F. GESZTESY AND M. WAURICK
We refer to [95, Ch. 5] for a detailed discussion of supersymmetric Dirac-typeoperators and the many explicit examples they represent.Next, we strengthen the hypotheses on Φ to the effect thatΦ ∈ C b (cid:0) R n ; C d × d (cid:1) , d ∈ N , (1.9)Φ( x ) = Φ( x ) ∗ , x ∈ R n , (1.10)there exists c > R > | Φ( x ) | > cI d , x ∈ R n \ B (0 , R ) , (1.11)and there exists ε > / α ∈ N n , | α | <
3, there is κ > k ( ∂ α Φ)( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | = 2 , x ∈ R n . (1.12) Theorem 1.1.
Let n ∈ N odd, n > . Under assumptions (1.9) – (1.12) on Φ ,the closed operator L := Q + Φ in L ( R n ) b n d is Fredholm with index given by theformula ind( L ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X i ,...,i n =1 ε i ...i n (1.13) × ˆ Λ S n − tr C d ( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x i n d n − σ ( x ) , where U ( x ) := | Φ( x ) | − Φ( x ) = sgn(Φ( x )) , x ∈ R n . Here ε i ··· i n denotes the totally anti-symmetric symbol in n coordinates, tr C d ( · )represents the matrix trace in C d × d , d n − σ ( · ) is the surface measure on the unitsphere S n − of R n , and we assumed n ∈ N to be odd since for algebraic reasons L has vanishing Fredholm index in all even spatial dimensions n (cf. (1.20) below).Theorem 1.1 represents the principal result of this manuscript and under thesehypotheses on Φ it is new as we suppose no additional asymptotic homogeneityproperties on Φ. In particular, it extends the original Callias formula for the indexof L to the hypotheses (1.9)–(1.12) on Φ. We also note that at the end of thismanuscript we take some first steps towards computing the Witten index of theoperator L under certain conditions on Φ in which L ceases to be Fredholm, yet itsWitten index is still given by a formula analogous to (1.13).For the topological setting underlying the Callias index formula (1.13) we referto the discussion by Bott and Seeley [14].Next, we succinctly summarize the principal strategy of proof underlying for-mula (1.13). While at first we follow Callias’ original strategy of proof, the bulk ofour arguments necessarily differ sharply from those in [22] as some of the claims in[22] can clearly be disproved (see our subsequent discussion). Step (1): Computing Fredholm indices abstractly.
Let H be a separableHilbert space, m ∈ N , and T ∈ B ( H m , H m ). Define the internal trace , tr m ( T ), of T by tr m ( T ) := m X j =1 T jj . (1.14) HE CALLIAS INDEX FORMULA REVISITED 5
Next, let M be a densely defined, closed linear operator in H m , and introduce theabbreviation B M ( z ) := z tr m (cid:0) ( M ∗ M + z ) − − ( M M ∗ + z ) − (cid:1) , z ∈ ̺ ( − M ∗ M ) ∩ ̺ ( − M M ∗ ) . (1.15)A basic result we employ to compute Fredholm indices then reads as follows: Theorem 1.2.
Assume that M is a densely defined, closed, and linear opera-tor in H m , and suppose that M is Fredholm. In addition, let { T Λ } Λ ∈ N , { S ∗ Λ } Λ ∈ N be sequences in B ( H ) , both strongly converging to I H as Λ → ∞ , and introduce S Λ := S ∗∗ Λ , Λ ∈ N . Assume that for each Λ ∈ N , there exists δ Λ > with Ω Λ := B (0 , δ Λ ) \{ } ⊆ ̺ ( − M M ∗ ) ∩ ̺ ( − M ∗ M ) and that the map Ω Λ ∋ z T Λ B M ( z ) S Λ (1.16) takes on values in B ( H ) , such that Ω Λ ∋ z tr H ( | T Λ B M ( z ) S Λ | ) = k T Λ B M ( z ) S Λ k B ( H ) is bounded ( w.r.t. z ) , (1.17) where tr H ( · ) represents the trace on B ( H ) , the Schatten-von Neumann ideal oftrace class operators on H . Then, ind( M ) = lim Λ →∞ lim z → tr H ( T Λ B M ( z ) S Λ ) . (1.18) In addition, if δ := 2 − inf Λ ∈ N ( δ Λ ) > and Ω := B (0 , δ ) ∋ z tr H ( T Λ B M ( z ) S Λ ) converges uniformly on B (0 , δ ) to some function F ( · ) as Λ → ∞ . Then, one caninterchange the limits Λ → ∞ and z → in (1.18) and obtains, F (0) = ind( M ) . (1.19)We emphasize that (1.18) and (1.19) represent a subtle, but crucial, deviationfrom the far simpler strategy employed in [22, Lemma 1] which entirely dispenseswith the additional regularization factors S Λ and T Λ , Λ ∈ N . At this point wedo not know if [22, Lemma 1] is valid, however, its proof is clearly invalid andwe record a counterexample (kindly communicated to us by H. Vogt [98]) to thestatement made on line 5 on p. 219 in the proof of [22, Lemma 1] later in Remark3.5 ( i ). After completing this project we became aware of an unpublished preprintby Arai [8] in which it was observed that the index regularization employed in [22]was insufficient. Step (2): Applying Step (1) to the operator L . One now identifies H and L ( R n ), m and 2 b n d , M and L , T Λ and the operator of multiplication by the char-acteristic function of the ball B (0 , Λ) ⊂ R n in L ( R n ), denoted by χ Λ , and chooses S ∗ Λ = I L ( R n ) , Λ ∈ N .According to (1.18) and especially, (1.19), we are thus interested in computingthe limit for Λ → ∞ of tr( χ Λ B L ( z )). Without loss of generality we restrict ourselvesin the following to n ∈ N odd, as a detailed analysis shows that actually B L ( z ) = 0 for n ∈ N , n even. (1.20)For z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) with Re( z ) > −
1, and n ∈ N odd, n >
3, one thenproceeds to prove that χ Λ B L ( z ) ∈ B (cid:0) L ( R n ) (cid:1) , and that the limit f ( z ) := lim Λ →∞ tr L ( R n ) ( χ Λ B L ( z ))exists. F. GESZTESY AND M. WAURICK
Step (3): Explicitly compute f ( z ) . A careful (and rather lengthy) evaluation of f ( z ) yields f ( z ) = (1 + z ) − n/ (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X i ,...,i n =1 ε i ...i n × ˆ Λ S n − tr C d ( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x i n d n − σ ( x ) , (1.21) z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , Re( z ) > − . However, at first we are only able to verify (1.21) for Re( z ) sufficienly large (asa consequence of relying on Neumann series expansions for resolvents). In order toderive (1.21) also for z in a neighborhood of 0, considerable additional efforts arerequired.Indeed, for achieving the existence of the limit Λ → ∞ in (1.21) for z in aneighborhood of 0, we employ Montel’s theorem and hence need to show that thefamily of analytic functions { z tr( χ Λ B L ( z )) } Λ constitutes a locally boundedfamily, that is, one needs to show that for all compact Ω ⊂ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ), sup Λ > sup z ∈ Ω | tr( χ Λ B L ( z )) | < ∞ . After proving local boundedness, we use Montel’s theorem for deducing that at leastfor a sequence { Λ k } k ∈ N with Λ k −→ k →∞ ∞ , the limit f := lim k →∞ tr( χ Λ k B L ( · )) existsin the compact open topology (i.e., the topology of uniform convergence on com-pacts). The explicit expression (1.21) for f then follows by the principle of analyticcontinuation and so carries over to z in a neighborhood of 0. In particular, since thelimit lim Λ →∞ tr( χ Λ B L (0)) exists and coincides with the index of L , we can thendeduce that independently of the sequence { Λ k } k ∈ N , the limit lim Λ →∞ tr( χ Λ B L ( · ))exists in the compact open topology and coincides with f given in (1.21). Thus, f (0) = ind( L )yields formula (1.13).We also emphasize that in connection with Steps (1)–(3), we perform thesecalculations only in the special case of admissible or τ -admissible potentials Φ(cf. Definitions 6.11 and 12.5) and then reduce the general case to τ -admissiblepotentials.It is clear from this short outline of our strategy of proof of Callias’ index formula(1.13), that in the end, our proof requires a fair number of additional steps notpresent in [22].Without entering any details at this point, we mention that one needs to dis-tinguish the case n = 3 from n > n = 3 due to the lack of regularity of certain integral kernels. In thiscontext we mention that it is unclear to us how continuity of the integral kernel of J iz on the diagonal, as claimed in [22, p. 224, line 6 from below], can be proved.Given our detailed approach, the number of resolvents applied is simply not largeenough to conclude continuity (see, in particular, Section 7).Perhaps, more drastically, trace class properties of certain integral operators aremerely dealt with by checking integrability of the integral kernel on the diagonal,see, for instance, the proof of [22, Lemma 5, p. 225]. HE CALLIAS INDEX FORMULA REVISITED 7
In addition, the claim that the expression X i ,...,i n ε i ...i n tr (( ∂ i Φ)( x ) . . . ( ∂ i n Φ)( x )) = 0 , x ∈ R n , (1.22)vanishes identically, is made on [22, p. 226]. A simple counter example can (locally)be constructed by demanding that Φ : R → C × is bounded, Φ ∈ C ∞ (cid:0) R ; C × (cid:1) ,and such that for one particular x ∈ R ,( ∂ Φ)( x ) = (cid:18) (cid:19) , ( ∂ Φ)( x ) = (cid:18) − (cid:19) , ( ∂ Φ)( x ) = (cid:18) i − i (cid:19) . In this case one verifies that X i ,i ,i ε i i i tr (( ∂ i Φ)( x )( ∂ i Φ)( x )( ∂ i Φ)( x )) = 24 i. These shortcomings in the arguments presented in [22] not withstanding, Cal-lias’ formula (1.13) is remarkable for its simplicity, as has been pointed out beforeby various authors. In particular, it is simpler, yet consistent with the Fedosov–H¨ormander formula [42], [43], [44], [45], [66], [67, Sect. 19.3] (derived with the helpof the pseudo-differential operator calculus), as discussed, for instance, in [3], [14],[91]. More precisely, the Fedosov–H¨ormander formula reads as follows,ind( L ) = − (cid:18) i π (cid:19) n ( n − n − ˆ ∂B tr (cid:16)(cid:0) σ − L d σ L (cid:1) ∧ (2 n − (cid:17) . (1.23)Here σ L : R n × R n → C b n d × b n d is the symbol of L given by σ L ( ξ, x ) = n X j =1 γ j,n iξ ⊗ I b n + I d ⊗ Φ( x ) , ξ, x ∈ R n ,B ⊆ R n is a ball of sufficiently large radius centered at the origin such that σ L isinvertible outside B , the orientation of R n × R n is given by dx ∧ dξ ∧· · ·∧ dx n ∧ dξ n >
0, and (cid:0) σ − L d σ L (cid:1) ∧ (2 n − is evaluated as a matrix product upon replacing ordinarymultiplication by the exterior product.The Callias index formula properly restated as the Fedosov–H¨ormander formulaand connections with half-bounded states were also discussed in [30]. Moreover, withthe help of the Cordes–Illner theory (see [36, 68] and the references in [85]), [85]established that the Fedosov–H¨ormander formula can also be used for computingthe index, if L is considered as an operator from the Sobolev space W ,p ( R n ) b n d to L p ( R n ) b n d for some p ∈ (1 , ∞ ). In addition, [86] (see also [87]) established thevalidity of the Fedosov–H¨ormander formula assuming the low regularity Φ ∈ C only (plus vanishing of derivatives at infinity).Callias employed Witten’s resolvent regularization inherent in (1.15), (1.18),(1.19), and we followed this device in this manuscript. For extensions to higherpowers of resolvents we refer to [94]. For connections between supersymmetricquantum mechanics, scattering theory and their connections with Witten’s resol-vent regularized index for Dirac-type operators in various space dimensions, andmatrix-valued (resp., operator-valued) coeffcients, we refer, for instance, to [6], [7],[12], [13], [20], [23], [24], [31], [75], [76], [79, Chs. IX, X], [80].The index problem for Dirac operators defined on complete Riemanniann man-ifolds has also been studied in [58] on the basis of relative index theorems (seealso [88]). Based on this approach, [4] found a generalized version of the Callias F. GESZTESY AND M. WAURICK index formula, which was further developed and connected with the Atiyah–Singerindex theorem in [5] (see also [28], [29], [41], [60] in this context). Independently,[89] found an alternative proof for the main result in [5], reducing the index prob-lem for the Dirac operator on a non-compact manifold to the compact case, thusmaking the index theorem in [10] applicable. Generalizing results in [4], and alsousing the Atiyah–Singer index theorem, [18] (see also [17]) derive index formulason manifolds, containing the Callias index formula as special case.For further generalizations of the index theorem for the Dirac operator to par-ticular manifolds, we refer to [48]. In addition, certain classes of Dirac operatorson even-dimensional manifolds are studied in [49], [47], [50], [51] employing K or KK -theory. The utility of KK -theory in view of the Callias index formula can alsobe seen in [74], where a short proof for the main results in [4] is given. Additionalconnections between K -theory and index theory for Dirac-type operators have beenestablished, for instance, in [21], [32], [33], [72], [73]. A rather different directionof index theory employing cyclic homology, aimed at even dimensional Dirac-typeoperators which generally are non-Fredholm, was undertaken in [25] (see also [26]).The approach to calculating Fredholm indices initiated by Callias [22] also had aprofound influence on theoretical physics as is amply demonstrated by the followingreferences [15], [34], [40], [46], [61], [62], [63], [64], [65], [69], [82], [83], [101], [102],[103], [104], and the literature cited therein.Returning to Theorem 1.1, we emphasize again that our derivation of the Calliasindex formula (1.13) under conditions (1.9)–(1.12) on Φ is new as the referencesjust mentioned either do not derive an explicit formula for ind( L ) in terms of Φ, orelse, derive the Fedosov–H¨ormander formula for ind( L ). All previous derivationsof (1.13) made some assumptions on Φ to the effect that asymptotically, Φ had tobe homogeneous of degree zero. We entirely dispensed with this condition in thismanuscript.We conclude this introduction with a brief description of the contents of eachsection. Our notational conventions are summarized in Section 2. Section 3 is de-voted to computing Fredholm indices employing Witten’s resolvent regularization.Schatten–von Neumann classes and trace class estimates are treated in Section 4.Pointwise bounds for integral kernels are developed in Section 5. The operator L underlying this manuscript is presented in Section 6. Trace class results, fundamen-tal for deriving formula (1.21) for f ( z ), are established in Section 7; estimates forintegral kernels on the diagonal and the computation of the trace of χ Λ B L ( z ) arediscussed in Section 8; the special case n = 3 is treated in Section 9. In Section 10we formulate our principal result, Theorem 10.2 (equivalently, Theorem 1.1) anddiscuss some consequences of formula (1.13). Perturbation theory of Helmholtzresolvents (Green’s functions) is isolated in Section 11. The proof of Theorem 10.2for smooth Φ is presented in Section 12; the case of general Φ satisfying (1.9)–(1.12)is concluded in Section 13. The final Section 14 is devoted to a particular class ofnon-Fredholm operators L and the associated Witten index. Appendix A presentsa concise construction of the Euclidean Dirac algebra, and Appendix B constructsan explicit counterexample to the trace class assertion in [22, Lemma 5]. Acknowledgments.
We thank Bernhelm Booß-Bavnbek, Alan Carey, StamatisDostoglou, Yuri Latushkin, Marius Mitrea, and Andreas Wipf for valuable discus-sions and/or correspondence. We are especially grateful to Alan Carey for extensivecorrespondence and hints to the literature, and to Stamatis Dostoglou for a critical
HE CALLIAS INDEX FORMULA REVISITED 9 reading of a substantial part of our manuscript and for many constructive commentsthat helped improving our exposition. We also thank Ralph Chill, Martin K¨orber,Rainer Picard, and Sascha Trostorff for useful discussions, particularly, concern-ing the counterexample discussed in Appendix B. In addition, we are indebted toHendrik Vogt for kindly communicating the example in Remark 3.5.M.W. gratefully acknowledges the hospitality of the Department of Mathematicsat the University of Missouri, USA, extended to him during two one-month visits inthe spring of 2014 and 2015. Moreover, he is particularly indebted to Yuri Tomilovin connection with the support received from the EU grant “AOS”, FP7-PEOPLE-2012-IRSES, No. 318910. Notational Conventions
For convenience of the reader we now summarize most of our notational conven-tions used throughout this manuscript.We find it convenient to employ the abbreviations, N > k := N ∩ [ k, ∞ ), k ∈ N , N = N ∪ { } , C Re >a := { z ∈ C | Re( z ) > a } , a ∈ R .The identity matrix in C r will be denoted by I r , r ∈ N .Let H be a separable complex Hilbert space, ( · , · ) H the scalar product in H (linear in the second factor), and I H the identity operator in H .Next, let T be a linear operator mapping (a subspace of) a Banach space intoanother, with dom( T ), ker( T ), and ran( T ) denoting the domain, kernel (i.e., nullspace), and range of T . The spectrum and resolvent set of a closed linear operatorin H will be denoted by σ ( · ) and ̺ ( · ). For resolvents of closed operators T acting ondom( T ) ⊆ H , we will frequently write ( T − z ) − rather than the precise ( T − zI H ) − , z ∈ ̺ ( T ).The Banach spaces of bounded and compact linear operators in H are denotedby B ( H ) and B ∞ ( H ), respectively. The Schatten–von Neumann ideals of compactlinear operators on H corresponding to ℓ p -summable singular values will be denotedby B p ( H ) or, if the Hilbert space under consideration is clear from the context (and,especially, for brevity in connection with proofs) just by B p , p ∈ [1 , ∞ ). The normson the respective spaces will be noted by k T k B p ( H ) for T ∈ B p ( H ), p ∈ [1 , ∞ ),and for ease of notation we will occasionally identify k T k B ( H ) with k T k B ∞ ( H ) for T ∈ B ( H ), but caution the reader that it is the set of compact operators on H that isdenoted by B ∞ ( H ). Similarly, B ( H , H ) and B ∞ ( H , H ) will be used for boundedand compact operators between two Hilbert spaces H and H . Moreover, X ֒ → X denotes the continuous embedding of the Banach space X into the Banach space X . Throughout this manuscript, if X denotes a Banach space, X ∗ denotes the adjoint space of continuous conjugate linear functionals on X , that is, the conjugatedual space of X (rather than the usual dual space of continuous linear functionalson X ). This avoids the well-known awkward distinction between adjoint operatorsin Banach and Hilbert spaces (cf., e.g., the pertinent discussion in [39, p. 3–4]). Inconnection with bounded linear functionals on X we will employ the usual bracketnotation h· , ·i X ∗ , X for pairings between elements of X ∗ and X .Whenever estimating the operator norm or a particular trace ideal norm of afinite product of operators, A A · · · A N , with A j ∈ B ( H ), j ∈ { , . . . , N } , N ∈ N ,we will simplify notation and write N Y j =1 A j , (2.1)disregarding any noncommutativity issues of the operators A j , j ∈ { , . . . , N } .This is of course permitted due to standard ideal properties and the associated(noncommutative) H¨older-type inequalities (see, e.g., [55, Sect. III.7], [92, Ch. 2]).The same convention will be applied if operators mapping between several Hilbertspaces are involved.We use the commutator symbol[ A, B ] := AB − BA (2.2)for suitable operators A, B . For unbounded A and B the natural domain of [ A, B ]is the intersection of the respective domains of AB and BA . In particular, [ A, B ] HE CALLIAS INDEX FORMULA REVISITED 11 is not closed in general. However, in the situations we are confronted with, weshall always be in the situation that [
A, B ] is densely defined and bounded, inparticular, it is closable with bounded closure. As this is always the case, we shall –in order to reduce a clumsy notation as much as possible – typically omit the closurebar (i.e., we use [
A, B ] rather than [
A, B ]). In fact, most of the operators underconsideration can be extended to suitable distribution spaces, such that seeminglyformal computations can be justified in the appropriate distribution space.w-lim and s-lim denote weak and strong limits in H as well as limits in the weakand strong operator topology for operators in B ( H ), n-lim denotes the norm limitof bounded operators operators in H (i.e., in the topology of B ( H )). C ∞ ( R n ) denotes the space of infinitely often differentiable functions with com-pact support in R n . We typically suppress the Lebesgue meausure in L p -spaces, L p ( R n ) := L p ( R n ; d n x ), k · k L p ( R n ; d n x ) := k · k p , and similarly, L p (Ω) := L p (Ω; d n x ),Ω ⊆ R n , p ∈ [1 , ∞ ) ∪ {∞} . To avoid too lengthy expressions, we will frequentlyjust write I rather than the precise I L ( R n ) , etc.Sometimes we use the symbol h· , ·i L ( R n ) (or, for brevity, especially in proofs,simply h· , ·i ), to indicate the fact that the scalar product ( · , · ) L ( R n ) in L ( R n ) hasbeen continuously extended to the pairing on the entire Sobolev scale, that is, weabreviate h· , ·i L ( R n ) := h· , ·i H − s ( R n ) ,H s ( R n ) , s > R n is denoted by S n − := { x ∈ R n | | x | = 1 } , with d n − σ ( · )representing the surface measure on S n − , n ∈ N > . The open ball in R n centeredat x ∈ R n of radius r > B ( x , r ).Since various matrix structures and tensor products are naturally associated withthe Dirac-type operators studied in this manuscript, we had to simplify the notationin several respects to avoid entirely unmanagably long expressions. For example,given d, b n ∈ N , spaces such as L ( R n ) ⊗ C d , L ( R n ) ⊗ C b n , and L ( R n ) ⊗ C b n ⊗ C d (and analogously for Sobolev spaces) will simply be denoted by L ( R n ) d , L ( R n ) b n ,and L ( R n ) b n d , respectively.In addition, given a d × d matrix Φ : R n → C d × d with entries given by boundedmeasurable functions, and given an element ψ ⊗ φ ∈ L ( R n ) b n ⊗ C d , we will fre-quently adhere to a slight abuse of notation and employ the symbol Φ also in thecontext of the operationΦ : ψ ⊗ φ ( x ψ ( x ) ⊗ Φ( x ) φ ) , (2.3)and acordingly then shorten this even further toΦ : ψ φ ( x ψ ( x )Φ( x ) φ ) , (2.4)Moreover, in connection with constant, invertible m × m matrices α ∈ C m × m andscalar differential expressions such as ∂ j , ∆, etc., we will use the notation α∂ j = ∂ j α, α ∆ = ∆ α (2.5)(with equality of domains) when applying these differential expressions to suffi-ciently regular functions of the type η ( · ) ⊗ c , c ∈ C m , abbreviated again by η ( · ) c .In the context of matrix-valued operators we also agree to use the followingnotational conventions: Given a scalar function f on R n , or a scalar linear operator R in L ( R n ), we will frequently identify f or R with the diagonal matrices f I m or R I m in L ( R n ) m × m for appropriate m ∈ N . Remark . We will identify a function Φ with its corresponding multiplicationoperator of multiplying by this function in a suitable function space. In doing so, fora differential operator Q , we will distinguish between the expression Q Φ and ( Q Φ)and, similarly, for other differential operators. Namely, Q Φ denotes the composition of the two operators Q and Φ, whereas ( Q Φ) denotes the multiplication operator of multplying by the function x ( Q Φ)( x ). ⋄ HE CALLIAS INDEX FORMULA REVISITED 13 Functional Analytic Preliminaries
In this section we shall summarize the results obtained by Callias in [22, Lemmas1 and 2]. We emphasize that we only succeeded to prove [22, Lemma 1] underthe stronger condition that the trace norm of the operator under consideration isbounded on a punctured neighborhood around the origin. To begin, we recall thesetting of [22, Section II, p. 218]:
Definition 3.1.
Let H be a separable Hilbert space, m ∈ N , and T ∈ B ( H m , H m ) ,a bounded linear operator from H m to H m . Denoting by ι j : H → H m the canonicalembedding defined by ι j h := { δ kj h } k ∈{ ,...,m } , we introduce T jk := ι ∗ j T ι k for j, k ∈{ , . . . , m } . We define the internal trace , tr m ( T ) , of T being the linear operator on H given by tr m ( T ) := m X j =1 T jj . (3.1) Next, let M be a densely defined, closed linear operator in H m and introduce Wit-ten’s resolvent regularization via B M ( z ) := z tr m (cid:0) ( M ∗ M + z ) − − ( M M ∗ + z ) − (cid:1) ∈ B ( H ) ,z ∈ ̺ ( − M ∗ M ) ∩ ̺ ( − M M ∗ ) . (3.2) We will denote by tr H ( · ) the trace on B ( H ) , the Schatten-von Neumann ideal oftrace class operators on H .Remark . Let H be a Hilbert space and let m ∈ N , T ∈ B ( H m ) and let T jk = ι ∗ j T ι k , j, k ∈ { , . . . , m } as in Definition 3.1. Then boundedness of ι ∗ j , ι k , j, k ∈ { , . . . , m } , and exploiting the ideal property of B ( H m ) yields T jk ∈ B ( H ) , j, k ∈ { , . . . , m } , in particular, tr m ( T ) ∈ B ( H ) . ⋄ It should be noted that in general, the internal trace does not satisfy the cyclicityproperty in the sense that for
A, B ∈ B ( H m , H m ),tr m ( AB ) = tr m ( BA ) . However, if one of the operators is actually a matrix with entries in C , then such aresult holds: Proposition 3.3.
Let H be a Hilbert space, m ∈ N , A ∈ B ( H m , H m ) , B ∈ C m × m . Then tr m ( AB ) = tr m ( BA ) . Proof.
We have A = ( A ij ) i,j ∈{ ,...,m } and B = ( B ij ) i,j ∈{ ,...,m } with A ij ∈ B ( H ) asin Definition 3.1 and B ij ∈ C . Then AB = (cid:18) X k ∈{ ,...,m } A ik B kj (cid:19) i,j ∈{ ,...,m } and BA = (cid:18) X k ∈{ ,...,m } B ik A kj (cid:19) i,j ∈{ ,...,m } . Hence, tr m ( AB ) = m X j =1 m X k =1 A jk B kj = m X j =1 m X k =1 B kj A jk = m X k =1 m X j =1 B kj A jk = tr m ( BA ) . (cid:3) Next, we need a result of the type of [22, Lemma 1], in fact, we need an addi-tional generalization of [22, Lemma 1] in order to be able to apply it to our situ-ation. We shall briefly recall the notions used in the next result: Given a Hilbertspace K , a Fredholm operator S : dom( S ) ⊆ K → K , denoted by S ∈ Φ( K ), is de-fined by S being a densely defined, closed, linear operator with finite-dimensionalnullspace, dim(ker( S )) < ∞ , and closed range, ran( S ), being finite-codimensional,dim(ker( S ∗ )) < ∞ . The Fredholm index , ind( S ), of a Fredholm operator S is thenthe difference of the dimension of the nullspace and codimension of the range, thatis, ind( S ) = dim(ker( S )) − dim(ker( S ∗ )) . Basic facts on Fredholm operators will be recalled at the end of this section. Forthe next lemma, we shall also use the notion of convergence in the strong operatortopology, that is, a sequence { T Λ } Λ ∈ N of bounded linear operators in a Hilbertspace H is said to converge to some T ∞ ∈ B ( H ) in the strong operator topology ,s-lim Λ →∞ T Λ = T ∞ , if for all φ ∈ H , we havelim Λ →∞ T Λ φ = T ∞ φ. Our (generalized) version of [22, Lemma 1] then reads as follows.
Theorem 3.4.
In the situation of Definition assume that M is Fredholm, andthat { T Λ } Λ ∈ N , { S ∗ Λ } Λ ∈ N are sequences in B ( H ) , both converging to I H in the strongoperator topology as λ → ∞ , and introduce S Λ := S ∗∗ Λ , Λ ∈ N . Let B M ( · ) be givenby (3.2) , B M ( z ) := z tr m (cid:0) ( M ∗ M + z ) − − ( M M ∗ + z ) − (cid:1) , z ∈ ̺ ( − M ∗ M ) ∩ ̺ ( − M M ∗ ) . (3.3) Assume that for each Λ ∈ N , there exists δ Λ > with Ω Λ := B (0 , δ Λ ) \{ } ⊆ ̺ ( − M M ∗ ) ∩ ̺ ( − M ∗ M ) and that the map Ω Λ ∋ z T Λ B M ( z ) S Λ takes on values in B ( H ) , such that Ω Λ ∋ z tr H ( | T Λ B M ( z ) S Λ | ) = k T Λ B M ( z ) S Λ k B ( H ) is bounded with respect to z . Then ind( M ) = lim Λ →∞ lim z → tr H ( T Λ B M ( z ) S Λ ) . (3.4) In addition, if δ := inf Λ ∈ N ( δ Λ ) > and Ω := B (0 , δ ) ∋ z tr H ( T Λ B M ( z ) S Λ ) converges uniformly on B (0 , δ ) to some function F ( · ) as Λ → ∞ . Then, one caninterchange the limits Λ → ∞ and z → in (3.4) and obtains, F (0) = ind( M ) . (3.5) HE CALLIAS INDEX FORMULA REVISITED 15
Proof.
By the Fredholm property of M , one deduces that M ∗ M and M M ∗ areFredholm and, if 0 lies in the spectrum of either M ∗ M or M M ∗ it is an isolatedeigenvalue of finite multiplicity. As M is Fredholm, ran( M ) is closed. Hence,ran( M ) = ker( M ∗ ) ⊥ , and since M is closed, ker( M ∗ M ) = ker( M ), as well as,ker( M M ∗ ) = ker( M ∗ ) . Denote by P ± : H m → H m the orthogonal projection ontoker( M ) and ker( M ∗ ), respectively. Since by hypothesis P ± are finite-dimensionaloperators, so is tr m ( P ± ). Moreover, we havetr H (tr m ( P ± )) = tr H m ( P ± ) = dim(ran( P ± )) . Indeed, the last equality being clear, we only need to show the first one. Let { φ k } k ∈ N be an orthonormal basis of H . Let ι j : H → H m be the canonical embed-ding given by ι j h := { δ ℓj h } ℓ ∈{ ,...,m } for all j ∈ { , . . . , m } . Then it is clear that { ι j φ k } j ∈{ ,...,m } ,k ∈ N constitutes an orthonormal basis for H m . We havetr H m ( P ± ) = m X j =1 X k ∈ N ( ι j φ k , P ± ι j φ k ) H m = m X j =1 X k ∈ N ( φ k , ι ∗ j P ± ι j φ k ) H = X k ∈ N m X j =1 ( φ k , P ± ,jj φ k ) H = X k ∈ N (cid:18) φ k , m X j =1 P ± ,jj φ k (cid:19) H = X k ∈ N ( φ k , tr m ( P ± ) φ k ) H = tr H (tr m ( P ± )) . Next, define for Λ ∈ N , Ω Λ ∋ z e B Λ ( z ) := T Λ (cid:2) tr m (cid:0) z ( M ∗ M + z ) − (cid:1) − tr m ( P + ) − tr m (cid:0) z ( M M ∗ + z ) − (cid:1) + tr m ( P − ) (cid:3) S Λ = T Λ B M ( z ) S Λ − T Λ tr m ( P + ) S Λ + T Λ tr m ( P − ) S Λ . By [71, Sect. III.6.5], z ( M ∗ M + z ) − − P + −→ z → z z ( M M ∗ + z ) − − P − . We note that e B Λ ( z ) ∈ B ( H ), z ∈ Ω Λ . Since Ω Λ ∋ z tr H ( | T Λ B M ( z ) S Λ | ) is bounded, so isΩ Λ ∋ z tr H (cid:0)(cid:12)(cid:12) e B Λ ( z ) (cid:12)(cid:12)(cid:1) . For the boundedness of Ω Λ ∋ z tr H (cid:0)(cid:12)(cid:12) e B Λ ( z ) (cid:12)(cid:12)(cid:1) itsuffices to observe thattr H (cid:0)(cid:12)(cid:12) e B Λ ( z ) (cid:12)(cid:12)(cid:1) = tr H (cid:0)(cid:12)(cid:12) T Λ B M ( z ) S Λ − T Λ tr m ( P + ) S Λ + T Λ tr m ( P − ) S Λ (cid:12)(cid:12)(cid:1) = (cid:13)(cid:13) T Λ B M ( z ) S Λ − T Λ tr m ( P + ) S Λ + T Λ tr m ( P − ) S Λ (cid:13)(cid:13) B ( H ) (cid:13)(cid:13) T Λ B M ( z ) S Λ (cid:13)(cid:13) B ( H ) + (cid:13)(cid:13) T Λ tr m ( P + ) S Λ (cid:13)(cid:13) B ( H ) + (cid:13)(cid:13) T Λ tr m ( P − ) S Λ (cid:13)(cid:13) B ( H ) = tr H (cid:0)(cid:12)(cid:12) T Λ B M ( z ) S Λ (cid:12)(cid:12)(cid:1) + tr H (cid:0)(cid:12)(cid:12) T Λ tr m ( P + ) S Λ (cid:12)(cid:12)(cid:1) + tr H (cid:0)(cid:12)(cid:12) T Λ tr m ( P − ) S Λ (cid:12)(cid:12)(cid:1) , z ∈ Ω Λ , and using that the last two summands correspond to traces of finite-rank oper-ators. Thus, from the analyticity of tr H (cid:0) e B Λ ( · ) F (cid:1) for every finite-rank operator F on H , one deduces that e B Λ ( · ) is analytic in the B ( H )-norm, see, for instance,[9, Proposition A.3] (or [100, Theorem A.4.3]). More precisely, in [100, TheoremA.4.3] there is the following characterization of analyticity of Banach space valuedfunctions: A function h : U → X for some open
U ⊆ C and some Banach space X is analytic if and only if U ∋ z
7→ k h ( z ) k X is bounded on compact subsets of U and z
7→ h h ( z ) , x ′ i is analytic for all x ′ ∈ V with V ⊆ X ′ being a norming set for X .Thus, it suffices to apply [100, Theorem A.4.3] to X = B ( H ) as underlying Banachspace, and to observe that the space of finite-rank operators forms a norming subsetof B ( H ) (cf. [9, Proposition A.3].By Riemann’s theorem on removable singularities, one deduces that e B Λ ( · ) isanalytic at 0 with respect to the B ( H )-norm. As e B Λ ( · ) is also norm analytic in B ( H ), and tends to 0 as z →
0, one gets that e B Λ ( z ) tends to 0 as z → B ( H )-norm. Hence,lim z → tr H ( T Λ B M ( z ) S Λ ) = lim z → (cid:16) tr H (cid:0) e B Λ ( z ) (cid:1) + tr H ( T Λ tr m ( P + ) S Λ ) − tr H ( T Λ tr m ( P − ) S Λ ) (cid:17) = 0 + tr H ( T Λ tr m ( P + ) S Λ ) − tr H ( T Λ tr m ( P − ) S Λ ) . Since s-lim Λ →∞ T Λ , S ∗ Λ = I H , one obtains T Λ tr m ( P ± ) S Λ −→ Λ →∞ tr m ( P ± ) in B ( H )(see, e.g., [105, Lemma 6.1.3]). Thus,lim Λ →∞ tr H ( T Λ tr m ( P ± ) S Λ ) = tr H (tr m ( P ± )) = tr H ( P ± ) . Hence, lim Λ →∞ lim z → tr H ( T Λ B M ( z ) S Λ )= 0 + lim Λ →∞ (cid:16) tr H ( T Λ tr m ( P + ) S Λ ) − tr H ( T Λ tr m ( P − ) S Λ ) (cid:17) = dim(ran( P + )) − dim(ran( P − ))= dim(ker( M )) − dim(ker( M ∗ ))= ind( M ) . Finally, for the purpose of proving the last statement of the theorem, define F Λ : Ω ∋ z tr H ( T Λ B M ( z ) S Λ ). Since { F Λ } Λ converges uniformly to F , one infers that F is continuous on Ω. Thus,ind( M ) = lim Λ →∞ lim z → F Λ ( z ) = lim Λ →∞ F Λ (0) = F (0) = lim z → F ( z ) = lim z → lim Λ →∞ F Λ ( z ) . (cid:3) In connection with the last part of Theorem 3.4 we note that the (limit of the)map z F ( z ) in 0 may be regarded as a generalized Witten index (see, e.g., [12],[53] and the references therein, as well as Section 14).
Remark . ( i ) While [22, p. 218, Lemma 1] might be valid as stated, it remainsunclear, how the assertion that is stated in line 5 on page 219 comes about. Theauthor infers the following: Let H be a separable Hilbert space, Γ ⊆ C openwith 0 ∈ ∂ Γ, B : Γ → B ( H ) analytic. Assume that B ( z ) ∈ B ( H ) for all z ∈ Γ,Γ ∋ z
7→ k B ( z ) k B ( H ) is bounded and k B ( z ) k B ( H ) → z →
0. Then for anorthonormal basis { φ k } k ∈ N of H ,tr H ( B ( z )) = ∞ X k =1 ( φ k , B ( z ) φ k ) H −→ z → . (3.6)This statement is invalid as the following example, kindly communicated to us by H.Vogt [98], shows: For the orthonormal basis { φ k } k ∈ N define the family of operators HE CALLIAS INDEX FORMULA REVISITED 17 B ( · ) by B ( z ) φ k := ze − ( k − z φ k , z ∈ Γ := { z ∈ C | | arg( z ) | < ( π/ , | z | < } , k ∈ N . Then k B ( z ) k B ( H ) = tr H ( | B ( z ) | ) = ∞ X k =1 ( φ k , | B ( z ) | φ k ) H = ∞ X k =1 | z | e − ( k − z ) = | z | ∞ X k =0 (cid:16) e − Re( z ) (cid:17) k = | z | − e − Re( z ) remains bounded for z ∈ Γ. Moreover, n-lim z → B ( z ) = 0 in B ( H ). However,tr H ( B ( z )) = ∞ X k =1 (cid:0) φ k , ze − ( k − z φ k (cid:1) H = z ∞ X k =0 e − kz = z − e − z = ze z − e z −→ z → . ( ii ) We shall now elaborate on the fact that an analytic function taking values inthe space of bounded linear operators in a Hilbert space can indeed have differentdomains of analyticity if considered as taking values in particular Schatten–vonNeumann ideals.Consider an infinite-dimensional Hilbert space H and pick an orthonormal basis { φ k } k ∈ N in H . For z ∈ C with Re( z ) >
0, define T ( z ) φ k := e − z ln( k ) φ k , k ∈ N . Then T ( z ) ∈ B ( H ) and T ( z ) ψ := ∞ X k =1 e − z ln( k ) ( φ k , ψ ) φ k , ψ ∈ H . Moreover, T (0) = I H , T (2) ∈ B ( H ), T (1) ∈ B ( H ). We note here that for Re( z ) > z T ( z ) is also analytic with values in B ( H ), however, the tracenorm of T ( · ) blows up at z = 1. ⋄ We conclude this section with some facts on Fredholm operators. For reasons tobe able to handle certain classes of unbounded Fredholm operators in a convenientmanner, we now take a slightly more general approach and permit a two-Hilbertspace setting as follows: Suppose H j , j ∈ { , } , are complex, separable Hilbertspaces. Then S : dom( S ) ⊆ H → H , S is called a Fredholm operator , denoted by S ∈ Φ( H , H ), if ( i ) S is closed and densely defined in H .( ii ) ran( S ) is closed in H .( iii ) dim(ker( S )) + dim(ker( S ∗ )) < ∞ .If S is Fredholm, its Fredholm index is given byind( S ) = dim(ker( S )) − dim(ker( S ∗ )) . (3.7)If S : dom( S ) ⊆ H → H is densely defined and closed, we associate withdom( S ) ⊂ H the standard graph Hilbert subspace H S ⊆ H induced by S definedby H S = (dom( S ); ( · , · ) H S ) , ( f, g ) H S = ( Sf, Sg ) H + ( f, g ) H , k f k H S = (cid:2) k Sf k H + k f k H (cid:3) / , f, g ∈ dom( S ) . In addition, for A , A ∈ Φ( H , H ) ∩B ( H , H ), A and A are called homotopic in Φ( H , H ) if there exists A : [0 , → B ( H , H ) continuous such that A ( t ) ∈ Φ( H , H ), t ∈ [0 , A (0) = A , A (1) = A .Next, following [11, Chs. 1, 3], [54, Chs. XI, XVII], [56, Sects. IV.6, IV.10], [77,Sect. I.3], [78, Ch. 2], [90, Chs. 5, 7], we now summarize a few basic properties ofFredholm operators. Theorem 3.6.
Let H j , j = 1 , , , be complex, separable Hilbert spaces, then thefollowing items ( i ) – ( vii ) hold: ( i ) If S ∈ Φ( H , H ) and T ∈ Φ( H , H ) , then T S ∈ Φ( H , H ) and ind( T S ) = ind( T ) + ind( S ) . (3.8)( ii ) Assume that S ∈ Φ( H , H ) and K ∈ B ∞ ( H , H ) , then ( S + K ) ∈ Φ( H , H ) and ind( S + K ) = ind( S ) . (3.9)( iii ) Suppose that S ∈ Φ( H , H ) and K ∈ B ∞ ( H S , H ) , then ( S + K ) ∈ Φ( H , H ) and ind( S + K ) = ind( S ) . (3.10)( iv ) Assume that S ∈ Φ( H , H ) . Then there exists ε ( S ) > such that for any R ∈ B ( H , H ) with k R k B ( H , H ) < ε ( S ) , one has ( S + R ) ∈ Φ( H , H ) and ind( S + R ) = ind( S ) , dim(ker( S + R )) dim(ker( S )) . (3.11)( v ) Let S ∈ Φ( H , H ) , then S ∗ ∈ Φ( H , H ) and ind( S ∗ ) = − ind( S ) . (3.12)( vi ) Assume that S ∈ Φ( H , H ) and that the Hilbert space V is continuously em-bedded in H , with dom( S ) dense in V . Then S ∈ Φ( V , H ) with ker( S ) and ran( S ) the same whether S is viewed as an operator S : dom( S ) ⊆ H → H , or asan operator S : dom( S ) ⊆ V → H . ( vii ) Assume that the Hilbert space W is continuously and densely embedded in H . If S ∈ Φ( W , H ) then S ∈ Φ( H , H ) with ker( S ) and ran( S ) the samewhether S is viewed as an operator S : dom( S ) ⊆ H → H , or as an operator S : dom( S ) ⊆ W → H . ( viii ) Homotopic operators in Φ( H , H ) ∩ B ( H , H ) have equal Fredholm in-dex. More precisely, the set Φ( H , H ) ∩ B ( H , H ) is open in B ( H , H ) , hence Φ( H , H ) contains at most countably many connected components, on each ofwhich the Fredholm index is constant. Equivalently, ind : Φ( H , H ) → Z is lo-cally constant, hence continuous, and homotopy invariant. A prime candidate for the Hilbert spaces V , W ⊆ H in Theorem 3.6 ( vi ), ( vii )(e.g., in applications to differential operators) is the graph Hilbert space H S inducedby S . Moreover, an immediate consequence of Theorem 3.6 we will apply later isthe following homotopy invariance of the Fredholm index for a family of Fredholmoperators with fixed domain. Corollary 3.7.
Let T ( s ) ∈ Φ( H , H ) , s ∈ I , where I ⊆ R is a connected interval,with dom( T ( s )) := V T independent of s ∈ I . In addition, assume that V T embedsdensely and continuously into H ( for instance, V T = H T ( s ) for some fixed s ∈ I ) and that T ( · ) is continuous with respect to the norm k · k B ( V T , H ) . Then ind( T ( s )) ∈ Z is independent of s ∈ I . (3.13) HE CALLIAS INDEX FORMULA REVISITED 19
The corresponding case of unbounded operators with varying domains (and H = H ) is treated in detail in [37]. On Schatten–von Neumann Classes and Trace Class Estimates
This is the first of two technical sections, providing basic results used later on inour detailed study of Dirac-type operators to be introduced in Section 6. We alsorecall results on the Schatten–von Neumann classes and apply these to concretesituations needed in Section 7.We start with the following well-known characterization of Hilbert–Schmidt op-erators B (cid:0) L (Ω; dµ ) (cid:1) in L (Ω; dµ ): Theorem 4.1 (see, e.g., [92, Theorem 2.11]) . Let (Ω; B ; µ ) be a separable measurespace and k : Ω × Ω → C be µ ⊗ µ measurable. Then the map U : ( L (Ω × Ω; dµ ⊗ dµ ) → B (cid:0) L (Ω; dµ ) (cid:1) ,k (cid:0) f (cid:0) Ω ∋ x ´ Ω k ( x, y ) f ( y ) dµ ( y ) (cid:1)(cid:1) , is unitary. The H¨older inequality is also valid for trace ideals with p -summable singularvalues. Theorem 4.2 (H¨older inequality, see, e.g., [92, Theorem 2.8]) . Assume that H isa complex, separable Hilbert space, m ∈ N , q j ∈ [1 , ∞ ] , j ∈ { , . . . , m } , p ∈ [1 , ∞ ] .Assume that m X j =1 q j = 1 p . Let T j ∈ B q j ( H ) , j ∈ { , . . . , m } . Then T := Q mj =1 T j ∈ B p ( H ) and k T k B p ( H ) m Y j =1 k T j k B qj ( H ) . For q = q = m = 2, one obtains a criterion for operators belonging to the traceclass B , which we shall use later on. Corollary 4.3.
Let (Ω; B , µ ) be a separable measure space, k : Ω × Ω → C be µ ⊗ µ measurable. Moreover, assume that there exists ℓ, m ∈ L (Ω × Ω; dµ ⊗ dµ ) such that k ( x, y ) = ˆ Ω ℓ ( x, w ) m ( w, y ) dµ ( w ) for µ ⊗ µ a.e. ( x, y ) ∈ Ω × Ω .Then K , the associated integral operator with integral kernel k ( · , · ) in L (Ω × Ω; dµ ⊗ dµ ) , is trace class, K ∈ B (cid:0) L (Ω; dµ ) (cid:1) , and tr L (Ω; dµ ) ( K ) = ˆ Ω k ( x, x ) dµ ( x ) = ˆ Ω ˆ Ω ℓ ( x, w ) m ( w, x ) dµ ( w ) dµ ( x ) . Proof.
By Theorem 4.1 the integral operators L and M associated with ℓ and m , respectively, are Hilbert–Schmidt operators. Since K = LM , one gets K ∈B (cid:0) L (Ω; dµ ) (cid:1) by Theorem 4.2. Moreover, by Theorem 4.1, one concludes thattr L (Ω; dµ ) ( K ) = tr L (Ω; dµ ) ( LM ) = tr L (Ω; dµ ) (cid:0) ( L ∗ ) ∗ M (cid:1) = ˆ Ω ˆ Ω ℓ ( w, x ) m ( x, w ) dµ ( x ) dµ ( w )= ˆ Ω ˆ Ω ℓ ( x, w ) m ( w, x ) dµ ( w ) dµ ( x ) . (cid:3) HE CALLIAS INDEX FORMULA REVISITED 21
In the bulk of this manuscript, Theorem 4.1 and Corollary 4.3 will be applied tothe case L (Ω; B ; dµ ) = L ( R n ; B ( R n ); d n x ) = L ( R n )We recall H ( R n ) and H ( R n ), the spaces of once and twice weakly differentiable L -functions with derivatives in L , respectively. Moreover, we shall furthermoreconsider the differential operator Q in L ( R n ) b n by Q := n X j =1 γ j,n ∂ j , dom( Q ) = H ( R n ) b n , (4.1)where γ ∗ j,n = γ j,n ∈ C b n × b n if n = 2 b n or n = 2 b n + 1, (4.2)and γ j,n γ k,n + γ k,n γ j,n = 2 δ jk for all j, k ∈ { , . . . , n } , see Definition A.3. A firstconsequence is, Q = ∆ I b n , dom( Q ) = H ( R n ) b n . (4.3)Indeed, QQ = n X j =1 γ j,n ∂ j n X k =1 γ k,n ∂ k = n X j =1 n X k =1 γ j,n ∂ j γ k,n ∂ k = 12 n X j =1 n X k =1 γ j,n ∂ j γ k,n ∂ k + 12 n X j =1 n X k =1 γ k,n ∂ k γ j,n ∂ j = 12 n X j =1 n X k =1 ( γ j,n γ k,n + γ k,n γ j,n ) ∂ j ∂ k = ∆ I b n . We study the operator asssociated with the differential expression Q with its prop-erties later on in Section 6. More precisely, in Theorem 6.4 we show that Q = − Q ∗ in L ( R n ) b n with domain H ( R n ) b n , that is, Q is skew-self-adjoint in L ( R n ) b n .In particular, we get for any µ ∈ C with Re( µ ) = 0 that (cid:13)(cid:13) ( Q + µ ) − (cid:13)(cid:13) | Re( µ ) | − (4.4)as an operator from L ( R n ) b n to L ( R n ) b n . One notes that by Fourier trans-form, the operator Q is unitarily equivalent to the Fourier multiplier with symbol P nj =1 γ j,n ( − i ) ξ j . Furthermore, by γ j,n = γ ∗ j,n and γ j,n = I b n , the matrix γ j,n isunitary. Hence, the symbol of Q may be estimated as follows (cid:13)(cid:13)(cid:13)(cid:13) n X j =1 γ j,n ( − i ) ξ j (cid:13)(cid:13)(cid:13)(cid:13) B ( C b n ) n X j =1 | ξ j | √ n (cid:18) n X j =1 | ξ j | (cid:19) / = √ n | ξ | , ξ ∈ R n . (4.5)We denote R µ := ( − ∆ + µ ) − , µ ∈ C \ ( −∞ , . (4.6)We recall our notational conventions collected in Section 2. In particular, werecall [ A, B ] = AB − BA , the commutator of two operators A and B , see also(6.15). Lemma 4.4.
Let µ ∈ C , Re( µ ) > , and Ψ ∈ C b ( R n ) . Then with Q and R µ givenby (4.1) and (4.6) , respectively, one obtains ( cf. Remark , [ R µ , Ψ] = R µ (cid:0) Q Ψ (cid:1) R µ + 2 R µ ( Q Ψ) QR µ . (4.7) Proof.
Recalling Remark 2.1 concerning multiplication operators, we compute withthe help of (4.3)[ R µ , Ψ] = R µ Ψ − Ψ R µ = R µ (Ψ ( − ∆ + µ ) − ( − ∆ + µ ) Ψ) R µ = R µ (∆Ψ − Ψ∆) R µ = R µ (cid:0) Q Ψ − Ψ∆ (cid:1) R µ = R µ ( Q ( Q Ψ) + Q Ψ Q − Ψ∆) R µ = R µ (cid:0)(cid:0) Q Ψ (cid:1) + ( Q Ψ) Q + ( Q Ψ) Q + Ψ Q − Ψ∆ (cid:1) R µ = R µ (cid:0)(cid:0) Q Ψ (cid:1) + 2 ( Q Ψ) Q (cid:1) R µ . (cid:3) In the course of computing the index of the closed operator L to be introducedlater on, we need to establish trace class properties of operators that are productsof operators of the form discussed in the following lemma. For given n ∈ N > , x ∈ R n , µ ∈ C Re > := { z ∈ C | Re( z ) > } , we let g µ ( x ) := 1Re( µ ) + | x | , e g µ ( x ) := √ n | x | Re( µ ) + | x | . (4.8)One notes that g µ ∈ L q ( R n ) for all q > n/ e g µ ∈ L q ( R n ) for all q > n . Lemma 4.5.
Let µ ∈ C Re > , Ψ ∈ L ∞ ( R n ) , α ∈ [1 , ∞ ) , n > , and recall R µ from (4.6) and Q from (4.1) , as well as g µ and e g µ from (4.8) . Assume that there exists κ > such that | Ψ( x ) | κ (1 + | x | ) − α for a.e. x ∈ R n . ( i ) Then for all q > n , R µ Ψ , Ψ R µ ∈ B q (cid:0) L ( R n ) (cid:1) and max (cid:0) k Ψ R µ k B q ( L ( R n )) , k R µ Ψ k B q ( L ( R n )) (cid:1) (2 π ) − n/q k Ψ k L q ( R n ) k g µ k L q ( R n ) < ∞ . The assertion remains the same, if R µ is replaced by R µ Q or QR µ and k g µ k L q ( R n ) in the latter estimate is replaced by k e g µ k L q ( R n ) . ( ii ) Assume, in addition, α > / . Then, if n > , there exists ϑ ∈ (3 / , suchthat R µ Ψ , Ψ R µ ∈ B nϑ/ (cid:0) L ( R n (cid:1) . Moreover, max (cid:0) k Ψ R µ k B nϑ/ ( L ( R n )) , k R µ Ψ k B nϑ/ ( L ( R n )) (cid:1) (2 π ) − / (2 ϑ ) k Ψ k L nϑ/ ( R n ) k g µ k L nϑ/ ( R n ) < ∞ . For n = 3 , R µ Ψ , Ψ R µ ∈ B (cid:0) L ( R n ) (cid:1) and max (cid:0) k Ψ R µ k B ( L ( R n )) , k R µ Ψ k B ( L ( R n )) (cid:1) (2 π ) − / k Ψ k L ( R n ) k g µ k L ( R n ) < ∞ . ( iii ) Let Θ ∈ C b ( R n ) with | ( Q Θ) ( x ) | + (cid:12)(cid:12)(cid:0) Q Θ (cid:1) ( x ) (cid:12)(cid:12) κ (1 + | x | ) − β for some κ > and β > / . Then, recalling (2.2) , [ R µ , Θ] ∈ B (cid:0) L ( R n ) (cid:1) with k [ R µ , Θ] k B ( L ( R n )) (cid:18) µ ) + 2 (cid:18) √ µ ) + 3 (cid:12)(cid:12) √ µ (cid:12)(cid:12) Re( µ ) (cid:19)(cid:19) (2 π ) − / × max (cid:0) k Q Θ k L ( R n ) , k Q Θ k L ( R n ) (cid:1) k g µ k L ( R n ) < ∞ HE CALLIAS INDEX FORMULA REVISITED 23 if n = 3 , and [ R µ , Θ] ∈ B (2 n/ ϑ (cid:0) L ( R n ) (cid:1) with k [ R µ , Θ] k B nϑ/ ( L ( R n )) (cid:18) µ ) + 2 (cid:18) √ µ ) + 3 (cid:12)(cid:12) √ µ (cid:12)(cid:12) Re( µ ) (cid:19)(cid:19) (2 π ) − / (2 ϑ ) × max (cid:0) k Q Θ k L nϑ/ ( R n ) , k Q Θ k L nϑ/ ( R n ) (cid:1) × k g µ k L nϑ/ ( R n ) < ∞ for some ϑ ∈ (0 , with nϑ/ > , if n > . The proof of Lemma 4.5, is basically contained in the following result:
Theorem 4.6 ([92, Theorem 4.1]) . Let n ∈ N , p > , and Ψ , g ∈ L p ( R n ) . De-fine T Ψ ,g ∈ B ( L ( R n )) as the operator of composition of multiplication by Ψ and g ( i∂ , . . . , i∂ n ) as a Fourier multiplier. Then T Ψ ,g ∈ B p ( L ( R n )) and k T Ψ ,g k B p ( L ( R n )) (2 π ) − n/p k Ψ k L p ( R n ) k g k L p ( R n ) . (4.9)The proof of (4.9) rests on the observation that there is equality for p = 2 anda straight forward estimate for the limiting case p = ∞ . The general case followsvia complex interpolation. Proof of Lemma . Observing k T k B p ( H ) = k T ∗ k B p ( H ) for all T ∈ B p ( H ), we shallonly show the respective assertions for Ψ R µ . For parts ( i ) and ( ii ) one uses Theorem4.6: One notes that for φ µ ( ξ ) := (cid:0) | ξ | + µ (cid:1) − , φ µ ( i∂ , . . . , i∂ n ) = R µ . Moreover,one observes that | φ µ | g µ ∈ L p ( R n ) for all p > n/
2. In order to prove item ( i ) onenotes that α > ∈ L q ( R n ) for all q > n . Hence, Ψ R µ ∈ B q (cid:0) L ( R n ) (cid:1) .The remaining assertion follows from the fact that the Fourier transform of QR µ liesin L q as it can be estimated by e g µ , see (4.5). For part ( ii ) one first considers the case n = 3. Then 2 α = α (2 / > n . Hence, Ψ ∈ L (cid:0) R (cid:1) and, since 2 > / n/ R µ ∈ B . If n >
3, there exists ϑ ∈ (3 / ,
1) such that αϑ > /
2. Inparticular, one has (2 / nϑ > (2 / /
4) = 2. Since α (2 / ϑn > (3 /
2) (2 n/
3) = n , one gets Ψ ∈ L nϑ/ ( R n ). Moreover, since (2 n/ ϑ > n/ ϑ > /
4, oneconcludes that | φ µ | g µ ∈ L nϑ/ ( R n ), implying Ψ R µ ∈ B nϑ/ . In order to showpart ( iii ) one notes that Lemma 4.4 implies[ R µ , Θ] = R µ (cid:0) Q Θ (cid:1) R µ + 2 R µ ( Q Θ) QR µ . Since QR µ is a bounded linear operator, using (4.3) as well as (4.6), one deducesfrom QR µ = ( Q + √ µ ) − ( Q + √ µ ) ( Q − √ µ ) R µ + √ µR µ = ( Q + √ µ ) − + √ µR µ its corresponding norm bound [Re( √ µ )] − + |√ µ | [Re( µ )] − , see (4.4) for the normbound of (cid:0) Q + √ µ (cid:1) − . Thus, the assertion follows from part ( ii ) and the idealproperty of the Schatten–von Neumann classes. (cid:3) Lemma 4.5 is decisive for obtaining the following result. We mention here thatH. Vogt [99] subsequently managed to prove the following theorem in a direct waywithout using Lemma 4.5 and thus without the use of Theorem 4.6. In the follow-ing theorem (and throughout this manuscript later on) we recall our simplifyingconvention (2.1) to abbreviate finite operator products A A · · · A N by Q Nj =1 A j , regardless of underlying noncommutativity issues, upon relying on ideal propertiesof the bounded operators A j , j = 1 , . . . , N , N ∈ N . Theorem 4.7.
Let n = 2 b n +1 ∈ N > odd, Ψ , . . . , Ψ b n +1 ∈ C b ( R n ) , α , . . . , α b n +1 ∈ [1 , ∞ ) , ε > / , κ > , µ ∈ C Re > , and let R µ , Q , and [Ψ b n , R µ ] be given by (4.6) , (4.1) , and (2.2) , respectively. ( i ) Assume that for all x ∈ R n and j ∈ { , . . . , b n + 1 } , | Ψ j ( x ) | κ (1 + | x | ) − α j . Then b n +1 Y j =1 Ψ j R µ , b n +1 Y j =1 R µ Ψ j ∈ B (cid:0) L ( R n ) (cid:1) , and (cid:13)(cid:13)(cid:13)(cid:13) b n +1 Y j =1 Ψ j R µ (cid:13)(cid:13)(cid:13)(cid:13) B ( L ( R n )) b n +1 Y j =1 k Ψ j R µ k B n +1 ( L ( R n )) . ( ii ) Assume for all x ∈ R n and j ∈ { , . . . , b n − } , | Ψ j ( x ) | κ (1 + | x | ) − α j , and | ( Q Ψ b n ) ( x ) | + (cid:12)(cid:12)(cid:0) Q Ψ b n (cid:1) ( x ) (cid:12)(cid:12) κ (1 + | x | ) − α b n − ε . Then b n − Y j =1 Ψ j R µ [Ψ b n , R µ ] ∈ B (cid:0) L ( R n ) (cid:1) . Proof.
In order to prove parts ( i ) and ( ii ), we use Theorem 4.2 and Lemma 4.5. Forpart ( i ) one observes that Ψ j R µ ∈ B n +1 by Lemma 4.5 ( i ) for all j ∈ { , . . . , b n + 1 } .Moreover, by P b n +1 j =1 1 n +1 = (cid:0) n − + 1 (cid:1) n +1 = 1 / Q b n +1 j =1 Ψ j R µ ∈ B .In order to arrive at item ( ii ), one notes that the case n = 3 directly follows fromLemma 4.5 ( iii ) since in that case b n − n > ϑ ∈ (3 / , b n , R µ ] ∈ B nϑ/ . The assertion is clear if 2 nϑ/
2. Thus, we assumethat 2 nϑ/ >
2. Let q ∈ R \{ } be such that( b n −
1) 1 q + 1(2 n/ ϑ = 12 . (4.10)Equation (4.10) with b n − n − / q = (cid:18) nϑ − nϑ (cid:19) n − (cid:18) nϑ − ϑnϑ (cid:19) n − (cid:18) ϑ − ϑ (cid:19) n ( n − < n . Hence, q > n and, so, from Ψ j R µ ∈ B q , by Lemma 4.5, the assertion follows fromTheorem 4.2. (cid:3) In order to illustrate the latter mechanism and for later purposes, we now discussan example.
HE CALLIAS INDEX FORMULA REVISITED 25
Example 4.8.
Let z > − , and Φ ∈ C ∞ b ( R ; C × ) such that for x ∈ R , with | x | > , Φ( x ) = X j =1 x j | x | σ j . Here σ := (cid:18) (cid:19) , σ := (cid:18) − ii (cid:19) , σ := (cid:18) − (cid:19) denote the Pauli matrices,see also Definition A.3 . Recalling our convention (2.5) and R z = ( − ∆+ 1 + z ) − ,we now prove that the operator given by T := tr (cid:18) X k =1 (cid:0) ( R z σ k ) ⊗ ( ∂ k Φ) (cid:1) × X k =1 (cid:0) ( R z σ k ) ⊗ ( ∂ k Φ) (cid:1) X k =1 (cid:0) ( R z σ k ) ⊗ ( ∂ k Φ) (cid:1) R z (cid:19) is trace class, T ∈ B (cid:0) L ( R ) (cid:1) .First of all, with the help of Proposition A.8 and introducing the fully anti-symmetric symbol in coordinates, ε jkℓ , j, k, ℓ ∈ { , , } , we may express T asfollows ( for notational simplicity, we now drop all tensor product symbols ) , T = X k ,k ,k tr (cid:0) σ k σ k σ k R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I (cid:1) = X k ,k ,k tr ( σ k σ k σ k ) × tr ( R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I )= X k ,k ,k iε k k k tr ( R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I ( ∂ k Φ) R z I )= X k ,k ,k iε k k k tr ( R z ( ∂ k Φ) R z ( ∂ k Φ) R z ( ∂ k Φ) R z )= X k ,k ,k iε k k k tr ( R z [( ∂ k Φ) , R z ]( ∂ k Φ) R z ( ∂ k Φ) R z )+ X k ,k ,k iε k k k tr ( R z R z ( ∂ k Φ)( ∂ k Φ) R z ( ∂ k Φ) R z )= X k ,k ,k iε k k k tr ( R z [( ∂ k Φ) , R z ]( ∂ k Φ) R z ( ∂ k Φ) R z )+ X k ,k ,k iε k k k tr ( R z R z ( ∂ k Φ)( ∂ k Φ)[ R z , ( ∂ k Φ)] R z )+ X k ,k ,k iε k k k tr ( R z R z ( ∂ k Φ)( ∂ k Φ)( ∂ k Φ) R z R z ) . One computes for k ∈ { , , } and x ∈ R , | x | > , ( ∂ k Φ)( x ) = X j =1 (cid:18) δ kj | x | − x j | x | x k | x | (cid:19) σ j , and observes that k ( ∂ k Φ)( x ) k / | x | . Moreover, it is easy to see that for all β ∈ N , with | β | := P j =1 β j > , there exists κ > such that for all x ∈ R n , k ( ∂ β Φ)( x ) k κ (1 + | x | ) − . The latter estimate together with Theorem yields that T ∈ B (cid:0) L ( R ) (cid:1) if andonly if e T := X k ,k ,k iε k k k tr ( R z R z ( ∂ k Φ)( ∂ k Φ)( ∂ k Φ) R z R z ) ∈ B (cid:0) L ( R ) (cid:1) . The latter operator can be rewritten as e T = X k ,k ,k iε k k k R z tr (( ∂ k Φ)( ∂ k Φ)( ∂ k Φ)) R z . Next, we inspect the term in the middle in more detail: X k ,k ,k ε k k k tr (( ∂ k Φ)( ∂ k Φ)( ∂ k Φ))= tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) − tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) + tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) − tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) + tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) − tr (( ∂ Φ)( ∂ Φ)( ∂ Φ))= 3 tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) − (( ∂ Φ)( ∂ Φ)( ∂ Φ)) . Employing ( ∂ Φ)( ∂ Φ)( ∂ Φ)( x )= X j ,j ,j (cid:18) δ j | x | − x j | x | x | x | (cid:19) σ j (cid:18) δ j | x | − x j | x | x | x | (cid:19) σ j (cid:18) δ j | x | − x j | x | x | x | (cid:19) σ j , one gets i | x | tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) ( x )= X j ,j ,j ε j j j (cid:18) δ j − x j x | x | (cid:19) (cid:18) δ j − x j x | x | (cid:19) (cid:18) δ j − x j x | x | (cid:19) = (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) − (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) + (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) − (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) + (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) − (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) (cid:18) − x x | x | (cid:19) = 1 − x | x | − x | x | − x | x | + x x | x | + x x | x | + x x | x | − x x x x | x | − x x x x | x | − x x x x | x | − x x x | x | + x x x | x | − x x x | x | + x x x | x | − x x x | x | + x x x | x | = 0 . On the other hand, i | x | tr (( ∂ Φ)( ∂ Φ)( ∂ Φ)) ( x )= X j ,j ,j ε j j j (cid:18) δ j − x j x | x | (cid:19) (cid:18) δ j − x j x | x | (cid:19) (cid:18) δ j − x j x | x | (cid:19) = 0 . We note that in this example, the corresponding formula (1.22) is in fact valid, for | x | > . This example is of a similar type as in [22, Section IV] . This may well bethe reason for the erroneous statement in [22, p. 226, 2nd highlighted formula] . We shall also use on occasion the following Hilbert–Schmidt criterion for exactly b n = ( n − / Theorem 4.9.
Let n = 2 b n + 1 ∈ N > odd, and assume that Ψ , . . . , Ψ b n ∈ L ∞ ( R n ) , α , . . . , α b n ∈ [1 , ∞ ) , µ ∈ C , Re( µ ) > . Let R µ , Q be given by (4.6) and (4.1) ,respectively. Assume that α j ∗ > / for some j ∗ ∈ { , . . . , b n } . Then T := (cid:18) j ∗ − Y j =1 Ψ j R µ (cid:19) Ψ j ∗ R µ (cid:18) b n Y j = j ∗ +1 Ψ j R µ (cid:19) ∈ B (cid:0) L ( R n ) (cid:1) , and k T k B ( L ( R n )) ((cid:0) Q j ∈{ ,..., b n }\{ j ∗ } k Ψ j R µ k B q ( L ( R n )) (cid:1) k Ψ ∗ j R µ k B r ( L ( R n )) , b n > , k Ψ R µ k B ( L ( R n )) , b n = 1 , where q = 2( b n − ϑ/ ( nϑ − > n and r = 2 nϑ/ > for some ϑ ∈ (3 / , ,according to Lemma ii ) .The assertion is the same if some of the factors with index j ∈ { , . . . , b n }\{ j ∗ } in the expression for T are replaced by Ψ j QR µ .Proof. By Lemma 4.5 ( ii ) one observes that for b n = j ∗ = 1, Ψ R µ ∈ B (cid:0) L ( R n ) (cid:1) ,and the assertion follows. The rest of the proof is similar to the one of the concludinglines of Theorem 4.7. (cid:3) Pointwise Estimates for Integral Kernels
The proof of the index theorem relies on (pointwise) estimates of integral kernelsof certain integral operators. These integral operators are of a form similar tothe one in Theorem 4.7. In order to guarantee that point-evaluation is a well-defined operation, these operators have to possess certain smoothing properties.Before proving the corresponding result, we define the Dirac δ -distribution of point-evaluation at some point x ∈ R n of a suitable function f : R n → C by δ { x } f := f ( x ) . We note that for every x ∈ R n one has δ { x } ∈ H − ( n/ − ε ( R n ) for all ε >
0, by theSobolev embedding theorem (see, e.g., [2, Theorem 7.34( c )]), and recall that H s ( R n ) := (cid:8) f ∈ L ( R n ) (cid:12)(cid:12) (1 + | · | ) s/ ( F f ) ∈ L ( R n ) (cid:9) , s ∈ R , (5.1)with norm denoted by k · k H s ( R n ) , where F denotes the (distributional) Fouriertransform being an extension of( F φ )( x ) := (2 π ) − n/ ˆ R n e − ixy φ ( y ) d n y, x ∈ R n , φ ∈ L ( R n ) ∩ L ( R n ) . (5.2)For f ∈ H n + ε ( R n ), we will find it convenient to write (cid:10) δ { x } , f (cid:11) L ( R n ) := δ { x } f = f ( x ) , (5.3)where h· , ·i L ( R n ) is understood as the continuous extension of the scalar producton L ( R n ) to the pairing on the entire fractional-order Sobolev scale, h· , ·i L ( R n ) := h· , ·i H − s ( R n ) ,H s ( R n ) , s > Theorem 5.1.
Let n ∈ N , ε > , T : H − ( n/ − ε ( R n ) → H ( n/ ε ( R n ) linear andbounded ( cf. (5.1) and (5.3)) . Then the map t : R n × R n ∋ ( x, y ) t ( x, y ) = (cid:10) δ { x } , T δ { y } (cid:11) L ( R n ) ∈ C is well-defined, continuous, and bounded. Moreover, if T ∈ B (cid:0) H − ( n/ − − ε ( R n ) , H ( n/ ε ( R n ) (cid:1) ∩ B (cid:0) H − ( n/ − ε ( R n ) , H ( n/ ε ( R n ) (cid:1) , (5.4) then t is bounded and continuously differentiable with bounded derivatives ∂ j t , j ∈{ , . . . , n } .Remark . We note that with the maps and assumptions introduced in Theorem5.1, t ( · , · ) is in fact the integral kernel of T , that is, t satisfies( T f )( x ) = ˆ R n t ( x, y ) f ( y ) d n y, x ∈ R n , for all f ∈ C ∞ ( R n ). Indeed, let x ∈ R n and f ∈ C ∞ ( R n ). Then one computes( T f )( x ) = (cid:10) δ { x } , T f (cid:11) L ( R n ) = (cid:10) T ∗ δ { x } , f (cid:11) L ( R n ) = ˆ R n T ∗ δ { x } ( y ) f ( y ) d n y = ˆ R n (cid:10) δ { y } , T ∗ δ { x } (cid:11) L ( R n ) f ( y ) d n y = ˆ R n (cid:10) T δ { y } , δ { x } (cid:11) L ( R n ) f ( y ) d n y = ˆ R n (cid:10) δ { x } , T δ { y } (cid:11) L ( R n ) f ( y ) d n y = ˆ R n t ( x, y ) f ( y ) d n y. HE CALLIAS INDEX FORMULA REVISITED 29 ⋄ Proof of Theorem . First, one observes that for any y ∈ R n , δ { y } ∈ H − n − ε ( R n )and F δ { y } ( x ) = (2 π ) − n/ e ixy for all x, y ∈ R n . Consequently, T δ { y } ∈ H n + ε ( R n ),and hence h δ { x } , T δ { y } i L ( R n ) is well-defined for all x, y ∈ R n . Moreover, from (cid:12)(cid:12) h δ { x } , T δ { y } i L ( R n ) (cid:12)(cid:12) (cid:12)(cid:12) δ { x } (cid:12)(cid:12) − n − ε (cid:12)(cid:12) T δ { y } (cid:12)(cid:12) n + ε (cid:12)(cid:12) δ { } (cid:12)(cid:12) − n − ε (cid:12)(cid:12) δ { } (cid:12)(cid:12) − n − ε × k T k B (cid:0) H − n − ε ( R n ) ,H n ε ( R n ) (cid:1) , x, y ∈ R n , one concludes the boundedness of t . Next, we show sequential continuity of t . Oneobserves that by the Sobolev embedding theorem, the map T δ { y } is continuous forall y ∈ R n (One recalls that F T δ { y } ∈ L ( R n ) with the Fourier transform given by(5.2).) Let { ( x k , y k ) } k ∈ N be a convergent sequence in R n × R n and denote its limitas k → ∞ by ( x, y ) . One notes that (cid:12)(cid:12) δ { y } − δ { y k } (cid:12)(cid:12) − n − ε → , as k → ∞ . Indeed,one gets by Lebesgue’s dominated convergence theorem that (cid:12)(cid:12) δ { y } − δ { y k } (cid:12)(cid:12) − n − ε = (2 π ) − n ˆ R n (cid:12)(cid:12)(cid:0) e ixy − e ixy k (cid:1)(cid:12)(cid:12) (cid:2) | x | (cid:3) − ( n/ − ε d n x −→ k →∞ . Moreover, one observes that { δ { x k } } k ∈ N is uniformly bounded in H − n − ε ( R n ) bysome constant M . Next, let η > k ∈ N such that for k > k one has (cid:12)(cid:12) δ { y } − δ { y k } (cid:12)(cid:12) − n − ε η and (cid:12)(cid:12) T δ { y } ( x k ) − T δ { y } ( x ) (cid:12)(cid:12) η . Then one estimates for k ∈ N , (cid:12)(cid:12) h δ { x k } , T δ { y k } i L ( R n ) − h δ { x } , T δ { y } i L ( R n ) (cid:12)(cid:12) (cid:12)(cid:12) h δ { x k } , T δ { y k } i L ( R n ) − h δ { x k } , T δ { y } i L ( R n ) (cid:12)(cid:12) + (cid:12)(cid:12) h δ { x k } , T δ { y } i L ( R n ) − h δ { x } , T δ { y } i L ( R n ) (cid:12)(cid:12) (cid:12)(cid:12) δ { x k } (cid:12)(cid:12) − n − ε (cid:12)(cid:12) T δ { y k } − T δ { y } (cid:12)(cid:12) n + ε + (cid:12)(cid:12) T δ { y } ( x k ) − T δ { y } ( x ) (cid:12)(cid:12) M k T k (cid:12)(cid:12) δ { y } − δ { y k } (cid:12)(cid:12) − n − ε + η ( M k T k + 1) η. Next, we turn to the second part of the theorem. Since H n + ε +1 ( R n ) ֒ → H n + ε ( R n ),the map t is continuous by the first part of the theorem. To prove differentiabil-ity, it suffices to observe that t has continuous (weak) partial derivatives. Since T ∂ j ∈ B (cid:0) H − n − ε ( R n ) , H n + ε ( R n ) (cid:1) and ∂ j T ∈ B (cid:0) H − n − ε ( R n ) , H n + ε ( R n ) (cid:1) for all j ∈ { , . . . , n } , the assertion also follows from the first part as one observes that( ∂ j t )( x, y ) = (cid:10) ( ∂ j δ ) { x } , T δ { y } (cid:11) L ( R n ) = − (cid:10) δ { x } , ∂ j T δ { y } (cid:11) L ( R n ) , ( ∂ j + n t )( x, y ) = h δ { x } , T ∂ j δ { y } i , j ∈ { , . . . , n } , ( x, y ) ∈ R n × R n . (cid:3) In the applications discussed later on, we shall be confronted with operatorsbeing already defined (or being extendable) to the whole fractional Sobolev scale.So the standard situation in which we will apply Theorem 5.1 is summarized in thefollowing corollary, with some examples in the succeeding proposition.
Corollary 5.3.
Let n ∈ N , k > n , S, T : S ℓ ∈ Z H ℓ ( R n ) → S ℓ ∈ Z H ℓ ( R n ) . Assumethat for all ℓ ∈ R , T ∈ B (cid:0) H ℓ ( R n ) , H ℓ + k ( R n ) (cid:1) and S ∈ B (cid:0) H ℓ ( R n ) , H ℓ + k +1 ( R n ) (cid:1) ,and introduce the maps t : R n × R n ∋ ( x, y ) (cid:10) δ { x } , T δ { y } (cid:11) L ( R n ) ,s : R n × R n ∋ ( x, y ) (cid:10) δ { x } , Sδ { y } (cid:11) L ( R n ) . Then t is bounded and continuous, and the map s is bounded and continuouslydifferentiable with bounded derivatives. ( See (5.1) and (5.3) for H s ( R n ) and δ { x } , x ∈ R n , s ∈ R . ) Proof.
This is a direct consequence of Theorem 5.1. (cid:3)
Proposition 5.4.
Let µ ∈ C , Re( µ ) > , ℓ ∈ R , and Φ ∈ C ∞ b ( R n ) . Then R µ =( − ∆ + µ ) − ∈ B (cid:0) H ℓ ( R n ) , H ℓ +2 ( R n ) (cid:1) and Φ ∈ B (cid:0) H ℓ ( R n ) (cid:1) .Proof. For ℓ ∈ Z , the first assertion follows easily with the help of the Fouriertransform, the second assertion is a straightforward induction argument for ℓ ∈ N ; for ℓ ∈ − N the result follows by duality. The results for ℓ ∈ R follow byinterpolation, see [96, Theorem 2.4.2]. (cid:3) The main issue of the considerations in this section are estimates of continu-ous integral kernels on the respective diagonals. An elementary estimate can beshown for integral operators which are induced by commutators with multiplicationoperators as the following result confirms.
Proposition 5.5.
Let n ∈ N , ε > , m ∈ N , and assume that T : H − ( n/ − ε ( R n ) → H ( n/ ε ( R n ) is linear and continuous, and that Φ ∈ C ∞ b ( R n ) . Then the map t Φ : R n × R n ∋ ( x, y ) (cid:10) δ { x } , [Φ , T ] δ { y } (cid:11) L ( R n ) ∈ C , where [Φ , T ] is given by (2.2) , is well-defined, continuous, bounded, and satisfies t Φ ( x, x ) = 0 , x ∈ R n .Proof. By Proposition 5.4 and Theorem 5.1, one gets that t Φ is well-defined, con-tinuous, and bounded. For x ∈ R n one then computes (cid:10) δ { x } , [Φ , T ] δ { y } (cid:11) = (cid:10) δ { x } , (Φ T − T Φ) δ { x } (cid:11) = (cid:10) δ { x } , Φ T δ { x } (cid:11) − (cid:10) δ { x } , ( T Φ) δ { x } (cid:11) = (cid:10) Φ ∗ δ { x } , T δ { x } (cid:11) − (cid:10) δ { x } , T (Φ δ ) { x } (cid:11) = (cid:10) Φ( x ) δ { x } , T δ { x } (cid:11) − (cid:10) δ { x } , T Φ( x ) δ { x } (cid:11) = (cid:10) δ { x } , Φ( x ) T δ { x } (cid:11) − (cid:10) δ { x } , T Φ( x ) δ { x } (cid:11) = 0 . (cid:3) The next lemma also discusses properties of the integral kernel of a commu-tator, however, in the following situation, we shall address the commutator withdifferentiation.
Lemma 5.6.
Let T ∈ B (cid:0) L ( R n ) (cid:1) be induced by the continuously differentiableintegral kernel t : R n × R n → C , j ∈ { , . . . , n } . If [ ∂ j , T ] ∈ B (cid:0) L ( R n ) (cid:1) , then [ ∂ j , T ] defined by (2.2) is an operator induced by the integral kernel ∂ j t + ∂ j + n t .Proof. Let x ∈ R n and f ∈ C ∞ ( R n ). One computes([ ∂ j , T ] f )( x ) = ( ∂ j T f )( x ) − ( T ∂ j f )( x )= ∂ j ˆ R n t ( x, y ) f ( y ) d n y − ˆ R n t ( x, y )( ∂ j f )( y ) d n y = ˆ R n ( ∂ j t )( x, y ) f ( y ) d n y + ˆ R n ( ∂ j + n t )( x, y ) f ( y ) d n y, We recall Remark 2.1: The symbol Φ is interpreted as the operator of multiplication by thefunction Φ. If ℓ <
0, this should read as multiplication in the distributional sense.
HE CALLIAS INDEX FORMULA REVISITED 31 using an integration by parts to arrive at the last equality. (cid:3)
Remark . We elaborate on an important consequence of Lemma 5.6 as follows:For j ∈ { , . . . , n } , let T j ∈ B ( L ( R n )) be induced by the continuously differentiableintegral kernel t j : R n × R n → C . Assume that [ ∂ j , T j ] ∈ B ( L ( R n )), j ∈ { , . . . , n } ,and consider the operator T := n X j =1 [ ∂ j , T j ] . By Lemma 5.6 one infers that the integral kernel t for T may be computed asfollows, t ( x, y ) = n X j =1 ( ∂ j t j + ∂ j + n t j )( x, y ) , x, y ∈ R n . Moreover, for g := { x t j ( x, x ) } j ∈{ ,...,n } , t ( x, x ) = n X j =1 ( ∂ j ( y t j ( y, y )))( x ) (5.5)= div ( g ( x )) , x ∈ R n . This observation will turn out to be useful when computing the trace of certainoperators. ⋄ The remaining section is devoted to obtaining pointwise estimates of variousintegral operators on the diagonal. For convenience, we recall the Γ-function (cf.[1, Sect. 6.1]), given byΓ( z ) := ˆ ∞ t z − e − t dt, z ∈ C Re > , as well as the n − S n − ⊆ R n , ω n − = 2 π ( n/ Γ (cid:0) n/ (cid:1) . (5.6) Proposition 5.8.
Let n, m ∈ N , m > ( n + 1) / , µ ∈ C , Re( µ ) > , and R µ , δ { } ,and Q be given by (4.6) , (5.3) , and (4.1) , respectively. Then (cid:12)(cid:12) R mµ δ { } (0) (cid:12)(cid:12) (cid:18) µ ) (cid:19) m (cid:16)p Re( µ ) (cid:17) n c, (5.7) (cid:12)(cid:12) QR mµ δ { } (0) (cid:12)(cid:12) (cid:18) µ ) (cid:19) m (cid:16)p Re( µ ) (cid:17) n +1 c ′ , (5.8) with c = (2 π ) − n ω n − ˆ ∞ r n − (cid:2) r + 1 (cid:3) − ( n +3) / dr, (5.9) and c ′ = (2 π ) − n √ nω n − ˆ ∞ r n (cid:2) r + 1 (cid:3) − ( n +3) / dr, (5.10) where ω n − is given by (5.6) . Proof.
We estimate R mµ δ { } (0) with the help of the Fourier transform as follows. (cid:12)(cid:12) R mµ δ { } (0) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:10) R mµ δ { } , δ { } (cid:11)(cid:12)(cid:12) = (2 π ) − n (cid:12)(cid:12)(cid:12)(cid:12) ˆ R n (cid:0) | ξ | + µ (cid:1) m d n ξ (cid:12)(cid:12)(cid:12)(cid:12) (2 π ) − n ˆ R n (cid:0) | ξ | + Re( µ ) (cid:1) m d n ξ = (2 π ) − n ω n − ˆ ∞ r n − ( r + Re( µ )) m dr = (2 π ) − n ω n − (cid:18) µ ) (cid:19) m ˆ ∞ r n − (cid:16)(cid:16) r √ Re( µ ) (cid:17) + 1 (cid:17) m dr = (2 π ) − n ω n − (cid:18) µ ) (cid:19) m ˆ ∞ (cid:0) t p Re( µ ) (cid:1) n − ( t + 1) m p Re( µ ) dt = (2 π ) − n ω n − (cid:18) µ ) (cid:19) m (cid:0)p Re( µ ) (cid:1) n ˆ ∞ t n − ( t + 1) m dt. In a similar fashion, one estimates (5.8), however, first we recall from (4.5) that (cid:12)(cid:12)(cid:12)P nj =1 γ j,n ( − i ) ξ j (cid:12)(cid:12)(cid:12) √ n | ξ | , ξ ∈ R n . Hence, one arrives at (cid:12)(cid:12) QR mµ δ { } (0) (cid:12)(cid:12) (2 π ) − n √ n ˆ R n | ξ | (cid:0) | ξ | + Re( µ ) (cid:1) m d n ξ = (2 π ) − n √ nω n − ˆ ∞ r n ( r + Re( µ )) m dr = (2 π ) − n √ nω n − (cid:18) µ ) (cid:19) m ˆ ∞ r n (cid:16)(cid:16) r √ Re( µ ) (cid:17) + 1 (cid:17) m dr = (2 π ) − n √ nω n − (cid:18) µ ) (cid:19) m (cid:0)p Re( µ ) (cid:1) n +1 ˆ ∞ t n ( t + 1) m dt. (cid:3) The main observation in this subsection, Lemma 5.14, needs some preparationswhich deal with the fundamental solution of the Helmholtz equation on R n for n > Lemma 5.9.
Let n, N ∈ N , and x , . . . , x N ∈ R n . Then | x | + N − X j =1 | x j +1 − x j | > max k N | x k | . Proof.
We proceed by induction. The case N = 1 is clear. For N ∈ N , one has | x | + N X j =1 | x j +1 − x j | > | x | + N X j =1 (cid:2) | x j +1 | − | x j | (cid:3) = | x N +1 | . Thus, employing the induction hypothesis, one gets that | x | + N X j =1 | x j +1 − x j | > max k N | x k | ∨ | x N +1 | = max ℓ N +1 | x ℓ | . (cid:3) HE CALLIAS INDEX FORMULA REVISITED 33
Lemma 5.10.
Let α > , β > . Then the map φ : [0 , ∞ ) ∋ r (cid:18) r (cid:19) α e − βr satisfies | φ ( r ) | ( [ α/ (2 β )] α e − α +2 β , α > β, , α β, r > . Proof.
From φ ′ ( r ) = (cid:18) α (cid:18) r (cid:19) α − − β (cid:18) r (cid:19) α (cid:19) e − βr = (cid:18) α − β (cid:18) r (cid:19)(cid:19) (cid:18) r (cid:19) α − e − βr one gets with r ∗ := ( α/β ) − φ ′ ( r ∗ ) = 0 if r ∗ >
0. Thus, by φ (0) = 1 and φ ( r ) → r → ∞ , one obtains the assertion. (cid:3) Next, we shall concentrate on pointwise estimates for the fundamental solution ofthe Helmholtz equation. We denote the integral kernel (i.e., the Helmholtz Green’sfunction) associated with ( − ∆ − z ) − by E n ( z ; x, y ), x, y ∈ R n , x = y , n ∈ N , n > z ∈ C . Then, E n ( z ; x, y )= ( i/ (cid:0) πz − / | x − y | (cid:1) (2 − n ) / H (1)( n − / (cid:0) z / | x − y | (cid:1) , n > , z ∈ C \{ } , − (2 π ) − ln( | x − y | ) , n = 2 , z = 0 , [( n − ω n − ] − | x − y | − n , n > , z = 0 , Im (cid:0) z / (cid:1) > , x, y ∈ R n , x = y, (5.11)where H (1) ν ( · ) denotes the Hankel function of the first kind with index ν > ω n − is given by (5.6). We will directly work with the explicitformula (5.11), even though one could also employ the Laplace transform connec-tion between the resolvent and the semigroup of − ∆ which manifests itself in theformula, E n ( z ; x, y ) = ˆ [0 , ∞ ) (4 πt ) − n/ e −| x − y | / (4 t ) e zt dt, Re( z ) < , x, y ∈ R n , x = y. (5.12)Later on, we need the following reformulation of the Helmholtz Green’s functionin odd space dimensions. We will use an explicit expression for the Hankel functionof the first kind. Lemma 5.11.
Let n = 2 b n + 1 ∈ N > odd, µ ∈ C Re > . We denote E n ( − z, r ) := E n ( z ; x, y ) , r > , z ∈ C \{ } , where x, y ∈ R n are such that | x − y | = r . Then the following formula holds E n ( µ, r ) = (cid:18) √ µ (cid:19) b n − (2 πr ) − b n e −√ µr b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µr (cid:19) k . Proof.
Our branch of √· is chosen such that √− z = i √ z for all z ∈ C with Re( z ) >
0. We use the following representation of the Hankel function of the first kind takenfrom [57, 8.466.1], H (1) b n − ( z ) = r πz i − b n e iz b n − X k =0 ( − k ( b n + k − k !( b n − k − iz ) k , b n ∈ N . Hence, E n ( µ, r ) = i (cid:18) πriµ / (cid:19) − b n H (1) b n − (cid:0) iµ / r (cid:1) = i (cid:18) πriµ / (cid:19) − b n s πiµ / r i − b n e iiµ / r b n − X k =0 ( − k ( b n + k − k !( b n − k − iiµ / r ) k = i (cid:18) πriµ / (cid:19) − b n s πiµ / r i − b n e − µ / r b n − X k =0 ( − k ( b n + k − k !( b n − k − − µ / r ) k = (cid:18) √ µ (cid:19) b n − (2 πr ) − b n e −√ µr b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µr (cid:19) k . (cid:3) As a first corollary to be drawn from the explicit formula in the latter result, wenow derive some estimates of the Helmholtz Green’s function.
Lemma 5.12.
Let n = 2 b n + 1 ∈ N > odd, µ ∈ C Re > . We denote E n ( µ, r ) := E n ( − µ ; x, y ) , r > , (5.13) with x, y ∈ R n such that | x − y | = r . Then the following assertions ( i ) – ( iii ) hold: ( i ) Assume that µ > , then for all r > , E n ( µ, r ) > . (5.14)( ii ) For all r > , |E n ( µ, r ) | (cid:16)p cos(arg( µ )) (cid:17) − b n E n (Re( µ ) , r ) . (5.15)( iii ) Assume that µ > , then for all r > , exp ( √ µr/ E n ( µ, r ) b n − E n ( µ/ , r ) . (5.16) Proof.
Assertion (5.14) is clear due to the fact that E n is the fundamental solutionof the positive, self-adjoint operator ( − ∆ + µ ). (Alternatively, one can also use theexplicit representation of E n ( · , · ) in Lemma 5.11.)In view of Lemma 5.11, in order to prove (5.15), it suffices to prove the followingtwo facts, |√ µ | p Re( µ ) = 1 p cos(arg( µ )) and | Re( √ µ ) | > p Re( µ ) . (5.17)To show these assertions, let ̺ > ϑ ∈ ( − π/ , π/
2) such that µ = ̺e iϑ . Then √ µ = √ ̺e iϑ/ = √ ̺ cos( ϑ/
2) + i √ ̺ sin( ϑ/
2) as well as p Re( µ ) = √ ̺ p cos( ϑ ).From p cos( ϑ ) = q (cos( ϑ/ − (sin( ϑ/ q (cos( ϑ/ = cos( ϑ/ , HE CALLIAS INDEX FORMULA REVISITED 35 and |√ µ | p Re( µ ) = |√ ̺e iϑ/ |√ ̺ p cos(arg( µ )) , assertion (5.17) follows.Finally, we turn to the proof of (5.16). Given the representation of E n ( · , · ) inLemma 5.11, one concludes thatexp (cid:18) √ µ r (cid:19) E n ( µ, r ) = exp (cid:18) √ µ r (cid:19) (cid:18) √ µ (cid:19) b n − (2 πr ) − b n e −√ µr × b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µr (cid:19) k = (cid:18) √ µ (cid:19) b n − (2 πr ) − b n e − √ µ r b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µr (cid:19) k b n − (cid:18) p µ/ (cid:19) b n − (2 πr ) − b n e − √ µ r b n − X k =0 ( b n + k − k !( b n − k − (cid:18) p µ r (cid:19) k . (cid:3) Next, we obtain similar results for the derivative of the fundamental solution.
Lemma 5.13.
Let n = 2 b n + 1 ∈ N > odd. Then, for all µ ∈ C Re > , there exists q µ : R > → R > with q µ ( | · | ) ∈ L ( R n ) , such that the following properties ( i ) – ( iii ) hold: ( i ) For all j ∈ { , . . . , n } and µ ∈ C , Re( µ ) > , and for all x, y ∈ R n , x = y , | ∂ j ( ξ E n ( − µ ; ξ, y )) ( x ) | q µ ( | x − y | ) . (5.18)( ii ) For all µ ∈ C , Re( µ ) > , q µ ( r ) (cid:16) / p cos(arg( µ )) (cid:17) b n q Re( µ ) ( r ) , r > . (5.19)( iii ) For all µ > , exp ( √ µr/ q µ ( r ) b n q µ/ ( r ) , r > . (5.20) Proof.
For r >
0, with E n ( µ, r ) as in Lemma 5.12 (and with the help of Lemma5.11), one obtains the following derivative of E n ( · , · ) with respect to the secondvariable,( ∂ r E n )( µ, r ) = − (cid:18) √ µ (cid:19) b n − (2 π ) − b n e −√ µr × b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µ (cid:19) k r − b n − k − ( √ µr + ( b n + k )) . Define for r > q µ ( r ) := (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) √ µ (cid:19) b n − (2 π ) − b n e −√ µr × b n − X k =0 ( b n + k − k !( b n − k − (cid:18) √ µ (cid:19) k r − b n − k − ( √ µr + ( b n + k )) (cid:12)(cid:12)(cid:12)(cid:12) . (5.21) Then q µ ( | · | ) ∈ L ( R n ). Indeed, due to the presence of the e −√ µr -term, onlyintegrability at x = 0 is an issue here. Since the order of the singularity of q µ ( | x | )at x = 0 is at most | x | − n since b n + ( b n −
1) + 1 = 2 b n < b n + 1 = n , also integrabilityof q µ ( | · | ) at x = 0 is ensured.To prove (5.18), one observes that for fixed y ∈ R n and y = x ∈ R n , | ∂ j ( ξ
7→ E n ( µ, | ξ − y | )) ( x ) | = q µ ( | x − y | ) (cid:12)(cid:12)(cid:12)(cid:12) x j − y j | x − y | (cid:12)(cid:12)(cid:12)(cid:12) q µ ( | x − y | ) . The assertion in (5.19) follows analogously to that of (5.15) with an explicit rep-resentation of √ µ , together with the observation that 1 / p cos(arg( µ )) >
1. Thesame arguments apply to the proof of (5.20). (cid:3)
Having established the preparations for estimating the integral kernel of productsof resolvents R µ of the Laplace operator and a multiplication operator Ψ j , we finallycome to the fundamental estimates (5.23) in Lemma 5.14. All the following resultswill be of a similar type. Namely, consider a productΨ R µ Ψ R µ · · · Ψ m R µ , (5.22)of smooth, bounded functions Ψ j , j ∈ { , . . . , m } , identified as multiplication op-erators in L ( R n ) and R µ = ( − ∆ + µ ) − for some µ ∈ C Re > . If m is sufficientlylarge (depending on the space dimension n ), the operator introduced in (5.22) hasa continuous integral kernel t : R n × R n → C . Roughly speaking, we will show thatthe behavior of x t ( x, x ) is determined by the (algebraic) decay properties of allthe functions Ψ j , j ∈ { , . . . , n } , that is, if Ψ j decays like | x | − α j for large | x | forsome α j > j ∈ { , . . . , m } , then x t ( x, x ) decays as | x | − P mj =1 α j . We will,however, need a more precise estimate. Namely, we also need to establish at thesame time the overall constant of this decay behavior as a function of µ . That iswhy we needed to establish Proposition 5.8, see also Remark 5.15 below.The precise statement regarding the estimate of the diagonal of such a continuousintegral kernel reads as follows: Lemma 5.14.
Let n = 2 b n + 1 ∈ N > , m > b n + 1 , and assume that Ψ , . . . , Ψ m +1 ∈ C ∞ b ( R n ) , and µ ∈ C , Re( µ ) > . Assume that there exists α j , κ j ∈ R > , j ∈{ , . . . , m + 1 } , such that | Ψ j ( x ) | κ j (1 + | x | ) − α j , x ∈ R n , j ∈ { , . . . , m + 1 } . Consider the integral kernels t and t k of T := Q mj =1 Ψ j R µ and T k := Q m +1 j =1 Ψ j,k R µ , respectively ( cf. Remark , where Ψ j,k = Ψ j (1 − δ jk ) + δ jk Ψ j Q , k ∈ { , . . . , n } ,with R µ and Q given by (4.1) and (4.6) , respectively. Then t and t k are continuousand there exists κ ′ > such that | t ( x, x ) | κ ′ [1 + ( | x | / − P mj =1 α j , | t k ( x, x ) | κ ′ [1 + ( | x | / − P m +1 j =1 α j , x ∈ R n . (5.23) For p Re( µ ) > P m +1 j =1 α j , one can choose, with c arg ( µ ) := cos(arg( µ )) − / , κ ′ = (cid:2) (2 c arg ( µ ) ) b n − (cid:3) m κ · · · κ m (cid:18) µ ) (cid:19) m (cid:18)r Re( µ )4 (cid:19) n c in the first estimate in (5.23) , and κ ′ = c arg ( µ ) (cid:2) ( c arg ( µ ) ) b n − (cid:3) m (2 b n ) m κ · · · κ m +1 (cid:18) µ ) (cid:19) m +1 (cid:18)r Re( µ )4 (cid:19) n +1 c ′ HE CALLIAS INDEX FORMULA REVISITED 37 in the second, with c and c ′ given by (5.9) and (5.10) , respectively.Proof. We shall only prove the assertion for T . The other assertions follow fromthe fact that the integral kernel of QR µ can be bounded by x c b n arg( µ ) µq Re( µ ) ( | x | ),see Lemma 5.13. Moreover, we shall exploit that the exponential estimates (5.16)and (5.19) in Lemmas 5.12 and 5.13, respectively, are essentially the same.The stated continuity of the integral kernels follows from 2 m > b n + 2 > n ,Corollary 5.3, and Proposition 5.4. Indeed, Proposition 5.4 implies that T ∈ L (cid:0) H ℓ ( R n ) , H ℓ +2 m ( R n ) (cid:1) , ℓ ∈ R . Thus, by Corollary 5.3, t : R n × R n : ( x, y ) (cid:10) δ { x } , T δ { y } (cid:11) is continuous as 2 m > n , with δ { x } , x ∈ R n , defined in (5.3).We denote the integral kernel of R µ = ( − ∆ + µ ) − by r µ . Then one notes that r µ ( x − y ) = E n ( − µ ; x, y ) = E n ( µ ; | x − y | ) . For simplicity, we now assume that µ is real (one recalls the estimate | r µ | c arg( µ ) r Re( µ ) with a positive real number c arg( µ ) depending on arg( µ ), see (5.15)).One observes that11 + | x + x |
11 + || x | − | x || = 11 + | x | − | x |
11 + | x | ,x, x ∈ R n , | x | > | x | . On the other hand, one obviousy also has11 + | x + x | , | x | | x | . Introducing the sets, B ( R ) := (cid:8) ( x , . . . .x m − ) ∈ ( R n ) m − (cid:12)(cid:12) max j m − | x j | R (cid:9) , ∁ B ( R ) := ( R n ) m − \ B ( R ) , R > , one computes for x ∈ R n , with e κ := κ · · · κ m +1 , | t ( x, x ) | = (cid:12)(cid:12)(cid:12)(cid:12) Ψ ( x ) ˆ ( R n ) m − r µ ( x − x )Ψ ( x ) r µ ( x − x ) · · · Ψ m ( x m − ) × r µ ( x m − − x ) d n x · · · d n x m − (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( R n ) m − | Ψ ( x ) | r µ ( x − x ) · · · | Ψ m ( x m − ) | r µ ( x m − − x ) × d n x · · · d n x m − = ˆ ( R n ) m − | Ψ ( x ) | r µ ( x ) · · · | Ψ m ( x m − + x ) | r µ ( x m − ) d n x · · · d n x m − e κ ˆ ( R n ) m − (cid:18)
11 + | x | (cid:19) α r µ ( x ) · · · (cid:18)
11 + | x m − + x | (cid:19) α m × r µ ( x m − ) d n x · · · d n x m − = e κ ˆ B ( | x | / (cid:18)
11 + | x | (cid:19) α r µ ( x ) · · · (cid:18)
11 + | x m − + x | (cid:19) α m × r µ ( x m − ) d n x · · · d n x m − + e κ ˆ ∁ B ( | x | / (cid:18)
11 + | x | (cid:19) α r µ ( x ) · · · (cid:18)
11 + | x m − + x | (cid:19) α m × r µ ( x m − ) d n x · · · d n x m − e κ (cid:0) | x | (cid:1) P mj =1 α j ˆ B ( | x | / r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − + e κ ˆ ∁ B ( | x | / r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) d n x · · · d n x m − e κ (cid:0) | x | (cid:1) P mj =1 α j ˆ ( R n ) m − r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − + e κ ˆ ∁ B ( | x | / r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) d n x · · · d n x m − . By Lemma 5.9 one recalls that for x , . . . , x m − ∈ R n , | x | + m − X j =1 | x j +1 − x j | + | x m − | > max j m − | x j | . With the latter observation one estimates, using (5.16), ˆ ∁ B ( | x | / r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) d n x · · · d n x m − = ˆ ∁ B ( | x | / e − √ µ ( | x | + P m − j =1 | x j +1 − x j | + | x m − | ) r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − (2 b n − ) m ˆ ∁ B ( | x | / e − √ µ ( max m − j =1 | x j | ) r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − (2 b n − ) m e − √ µ | x | ˆ ∁ B ( | x | / r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − (2 b n − ) m e − √ µ | x | ˆ ( R n ) m − r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) × d n x · · · d n x m − . The latter expression decays faster than any power of (1 + | x | ) − . In fact, givenLemma 5.10, for √ µ > P m +1 j =1 α j , one obtains e − √ µ | x | [1 + ( | x | / P m +1 j =1 α j κ ′ > | t ( x, x ) | (2 b n − ) m κ ′ [1 + ( | x | / − P mj =1 α j , x ∈ R n . For the more precise estimate, one observes that ˆ ( R n ) m − r µ ( x ) r µ ( x − x ) · · · r µ ( x m − ) d n x · · · d n x m − = R m µ δ { } (0) HE CALLIAS INDEX FORMULA REVISITED 39 and then applies Proposition 5.8 to estimate the latter expression. (cid:3)
Remark . A further inspection of Lemma 5.14 reveals that if p Re( µ ) > ℓ forsome ℓ P m +1 j =1 α j , one obtains the estimates | t ( x, x ) | (cid:2) (2 c arg µ ) b n − (cid:3) m κ · · · κ m (cid:18) µ ) (cid:19) m (cid:18)r Re( µ )4 (cid:19) n c [1 + ( | x | / − ℓ ,x ∈ R n , | t k ( x, x ) | c arg µ (cid:2) ( c arg µ ) b n − (cid:3) m b n m κ · · · κ m +1 ×× (cid:18) µ ) (cid:19) m +1 (cid:18)r Re( µ )4 (cid:19) n +1 c ′ [1 + ( | x | / − ℓ , x ∈ R n , with c and c ′ given by (5.9) and (5.10), respectively.To illustrate the importance of this result, envisage a product as in (5.22) with m factors, all of them decaying like | x | − as | x | → ∞ . Later on, we shall see that incertain integrals it suffices to estimate the diagonal decaying like | x | − n as | x | → ∞ .Thus, if m is fairly large compared to n , and hence Lemma 5.14 yields a decay like | x | − m , we have to choose the real-part of µ rather large as the explicit constant isonly valid for p Re( µ ) > m . But, if we are only interested in an estimate of thetype | x | − n , we may choose p Re( µ ) a apriori just larger than n . ⋄ A readily applicable version of Lemma 5.14 reads as follows.
Lemma 5.16.
Let n = 2 b n + 1 ∈ N > odd, m > b n + 1 , assume that Ψ , . . . , Ψ m ∈ C ∞ b ( R n ) , and suppose that µ ∈ C Re > . Let R µ and Q be given by (4.6) and (4.1) ,respectively. Assume that there exists α j , κ j ∈ R > , j ∈ { , . . . , m } , such that | Ψ j ( x ) | κ j (1 + | x | ) − α j , x ∈ R n , j ∈ { , . . . , m } . Let ℓ ∈ { , . . . , m } and assume that there exists ε > such that | ( Q Ψ ℓ ) ( x ) | + (cid:12)(cid:12)(cid:0) Q Ψ ℓ (cid:1) ( x ) (cid:12)(cid:12) κ ℓ (1 + | x | ) − α ℓ − ε , x ∈ R n . If t denotes the integral kernel of T := ℓ − Y j =1 (Ψ j R µ ) Ψ ℓ − [ R µ , Ψ ℓ ] R µ m Y j = ℓ +1 Ψ j R µ ( cf. Remark and (2.2)) , then t is continuous on the diagonal and there exists κ ′ > such that | t ( x, x ) | κ ′ [1 + ( | x | / − ε − P mj =1 α j , x ∈ R n . If p Re( µ ) > P mj =1 α j + 2 ε , then a possible choice for κ ′ is κ ′ = κ · · · κ m (cid:18) µ ) (cid:19) m (cid:18)r Re( µ )4 (cid:19) n d, (5.24) where d := (cid:2) (2 c arg µ ) b n − (cid:3) m c + 2 c b n arg µ (cid:2) ( c arg µ ) b n − (cid:3) m − b n m c ′ , with c and c ′ given by (5.9) and (5.10) , respectively.Proof. One recalls from Lemma 4.4 (see also Remark 2.1) that[ R µ , Ψ ℓ ] = R µ (cid:0) Q Ψ ℓ (cid:1) R µ + 2 R µ ( Q Ψ ℓ ) QR µ . Let t be the associated integral kernel ofΨ R µ · · · Ψ ℓ − R µ (cid:0) Q Ψ ℓ (cid:1) R µ R µ Ψ ℓ +1 R µ · · · Ψ m R µ and t the one ofΨ R µ · · · Ψ ℓ − R µ ( Q Ψ ℓ ) QR µ R µ Ψ ℓ +1 R µ · · · Ψ m R µ . By hypothesis and by Lemma 5.14, for some constant κ ′ > | t ( x, x ) | + | t ( x, x ) | κ ′ [1 + ( | x | / − ε − P mj =1 α j , x ∈ R n . (5.25)The quantitative version of this assertion (i.e., the fact that κ ′ given by (5.24) is apossible choice in the estimate (5.25)), also follows from Lemma 5.14. (cid:3) Finally, we state one more variant of Lemma 5.14.
Lemma 5.17.
Let n = 2 b n + 1 ∈ N > odd, m > b n + 1 , assume that Ψ , . . . , Ψ m ∈ C ∞ b ( R n ) , and µ ∈ C , Re( µ ) > . Let R µ be given by (4.6) , and Q by (4.1) . Let ε, α j , κ j ∈ R > , and assume that for all j ∈ { , . . . , m } and ℓ ∈ { , . . . , m } , | Ψ j ( x ) | κ j (1 + | x | ) − α j , | ( Q Ψ ℓ ) ( x ) | + (cid:12)(cid:12)(cid:0) Q Ψ ℓ (cid:1) ( x ) (cid:12)(cid:12) κ ℓ [1 + ( | x | / − α ℓ − ε ,x ∈ R n . Then for ℓ ∈ { , . . . , m } , the associated integral kernels h ℓ and e h ℓ of (cid:18) ℓ Y j =1 Ψ j R µ (cid:19)(cid:18) m Y j = ℓ +1 Ψ j (cid:19) R m − ℓµ and (cid:18) ℓ − Y j =1 Ψ j R µ (cid:19)(cid:18) m Y j = ℓ Ψ j (cid:19) R m − ℓ +1 µ , respectively ( cf. Remark , satisfy for some κ ′ > , (cid:12)(cid:12) h ℓ ( x, x ) − e h ℓ ( x, x ) (cid:12)(cid:12) κ ′ [1 + ( | x | / − ε − P mj =1 α j , x ∈ R n . In addition, if p Re( µ ) > P mj =1 α j + 2 ε , then a possible choice for κ ′ is given by (5.24) .Proof. We will exploit Lemma 5.16 and note that h ℓ − e h ℓ is the associated integralkernel of the operator (cid:18) ℓ Y j =1 Ψ j R µ (cid:19)(cid:18) m Y j = ℓ +1 Ψ j (cid:19) R m − ℓµ − (cid:18) ℓ − Y j =1 Ψ j R µ (cid:19)(cid:18) m Y j = ℓ Ψ j (cid:19) R m − ℓ +1 µ = (cid:18)(cid:18) ℓ − Y j =1 Ψ j R µ (cid:19) Ψ ℓ (cid:19)(cid:18) R µ (cid:18) m Y j = ℓ +1 Ψ j (cid:19) − (cid:18) m Y j = ℓ +1 Ψ j (cid:19) R µ (cid:19) R m − ℓµ = (cid:18)(cid:18) ℓ − Y j =1 Ψ j R µ (cid:19) Ψ ℓ (cid:19)(cid:18)(cid:20) R µ , (cid:18) m Y j = ℓ +1 Ψ j (cid:19)(cid:21)(cid:19) R m − ℓµ . By hypothesis, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) m Y j = ℓ +1 Ψ j (cid:19) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) κ ℓ +1 · · · κ m (1 + | x | ) − P mj = ℓ +1 α j , x ∈ R n . Moreover, by the product rule one concludes that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Q m Y j = ℓ +1 Ψ j (cid:19) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) κ ℓ +1 · · · κ m (1 + | x | ) − ε − P mj = ℓ +1 α j , x ∈ R n , HE CALLIAS INDEX FORMULA REVISITED 41 and thus also that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Q m Y j = ℓ +1 Ψ j (cid:19) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) κ ℓ +1 · · · κ m (1 + | x | ) − ε − P mj = ℓ +1 α j , x ∈ R n . Hence, the assertion indeed follows from Lemma 5.16. (cid:3)
Remark . Iterated application Lemma 5.17 shows that under the same assump-tions, the integral kernels h m and e h of (cid:18) m Y j =1 Ψ j R µ (cid:19) and (cid:18) m Y j =1 Ψ j (cid:19) R mµ , respectively, satisfy for some κ ′ > (cid:12)(cid:12) h m ( x, x ) − e h ( x, x ) (cid:12)(cid:12) κ ′ [1 + ( | x | / − ε − P mj =1 α j , x ∈ R n . ⋄ Dirac-Type Operators
In this section, we discuss the operator L with a bounded smooth potential,studied by Callias [22] in L ( R n ) p for a suitable p ∈ N . We compute its domain,its adjoint and give conditions for the Fredholm property of this operator.Let H j ( R n ) := (cid:8) f ∈ L ( R n ) (cid:12)(cid:12) ∂ j f ∈ L ( R n ) (cid:9) , j ∈ { , . . . , n } , where ∂ j f denotes the distributional partial derivative of f ∈ L ( R n ) with respectto the j th variable. One notes that (see also (5.1)) H ( R n ) = \ j ∈{ ,...,n } H j ( R n ) . Remark . In the following, we make use of the so-called
Euclidean Dirac algebra ,see Appendix A and Definition A.3 for the construction and some basic properties.For dimension n ∈ N we denote the elements of this algebra by γ j,n , j ∈ { , . . . , n } .One recalls that for n = 2 b n or n = 2 b n + 1 for some b n ∈ N one has γ ∗ j,n = γ j,n ∈ C b n × b n , γ j,n γ k,n + γ k,n γ j,n = 2 δ jk I b n , j, k ∈ { , . . . , n } . (6.1) ⋄ We are now in the position to properly define the operator L (and the underlyingsupersymmetric Dirac-type operator H ) to be studied in the rest of this manuscript. Definition 6.2.
Let d ∈ N and suppose that Φ : R n → C d × d is a bounded measur-able function assuming values in the space of d × d self-adjoint matrices. We recallour convention H ( R n ) b n d = H ( R n ) b n ⊗ C d . With this in mind, we introduce the ( closed ) operator L in L ( R n ) b n d via L : ( H ( R n ) b n d ⊆ L ( R n ) b n d → L ( R n ) b n d ,ψ ⊗ φ (cid:16)P nj =1 γ j,n ∂ j ψ (cid:17) ⊗ φ + ( x ψ ( x ) ⊗ Φ( x ) φ ) . (6.2) Henceforth, recalling (4.1) , we shall abreviate Q := Q ⊗ I d = (cid:18) n X j =1 γ j,n ∂ j (cid:19) I d , (6.3) and, with a slight abuse of notation, employ the symbol Φ also in the context of theoperation Φ : ψ ⊗ φ ( x ψ ( x ) ⊗ Φ( x ) φ ) , (6.4) see also our notational conventions to suppress tensor products whenever possible,collected in Section and in Remark . Thus, L = Q + Φ . (6.5) Finally, the underlying ( self-adjoint ) supersymmetric Dirac-type operator H in L ( R n ) b n d ⊕ L ( R n ) b n d is of the form H = (cid:18) L ∗ L (cid:19) . (6.6) HE CALLIAS INDEX FORMULA REVISITED 43
A detailed treatment of supersymmetric Dirac-type operators can be found in[95, Ch. 5].For clarity, we kept the tensor product notation in (6.2)–(6.4), but from now onwe will typically dispense with tensor products to simplify notation.In this section, we shall prove the following theorem:
Theorem 6.3 ([22, Corollary on p. 217]) . Consider the operator L given by (6.2) .Assume, in addition, that Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) , Φ( x ) = Φ( x ) ∗ , x ∈ R n , and assumethere exists c > such that | Φ( x ) | > cI d , x ∈ R n , as well as ( Q Φ) ( x ) → as | x | → ∞ . Then the operator L = Q + Φ , dom( L ) = dom( Q ) = H ( R n ) b n d , (6.7) is closed and Fredholm in L ( R n ) b n d . Consequently, the supersymmetric Dirac-typeoperator H = (cid:18) L ∗ L (cid:19) , dom( H ) = H ( R n ) b n d ⊕ H ( R n ) b n d , (6.8) is self-adjoint and Fredholm in L ( R n ) b n d ⊕ L ( R n ) b n d . In order to deduce Theorem 6.3, we need some preparations. The first result isconcerned with the operator Q in L ( R n ) b n given by (4.1). Moreover, we will showthat Q is skew-self-adjoint and thus verify the estimate asserted in (4.4). Theorem 6.4.
Let n ∈ N > and hence γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , with n = 2 b n or n = 2 b n + 1 , see Remark . Denote ∂ j : H j ( R n ) b n ⊆ L ( R n ) b n → L ( R n ) b n , f ∂ j f, j ∈ { , . . . , n } . Then the following assertions ( i ) – ( iii ) hold: ( i ) ∂ j is a skew-self-adjoint operator, j ∈ { , . . . , n } . ( ii ) γ j,n ∂ j = ∂ j γ j,n is skew-self-adjoint, j ∈ { , . . . , n } . ( iii ) Q = P nj =1 γ j,n ∂ j , dom( Q ) = H ( R n ) b n is skew-self-adjoint ( and thus closed ) in L ( R n ) b n .Proof. By Fourier transform (see (5.2)), the operator ∂ j is unitarily equivalent tothe operator given by multiplication by the function x ix j . The latter is amultiplication operator taking values on the imaginary axis; thus, it is skew-self-adjoint. Hence, so is ∂ j , proving assertion ( i ).Let j ∈ { , . . . , n } . Assertion ( ii ) follows from observing that γ j,n defines anisomorphism from L ( R n ) b n into itself. Indeed, this follows from the fact that γ j,n = I b n . Moreover, since γ j,n is a constant coefficient matrix it leaves the space C ∞ ( R n ) b n invariant. The equality γ j,n ∂ j φ = ∂ j γ j,n φ is clear for φ ∈ C ∞ ( R n ) b n .Hence, ∂ j γ j,n ⊇ γ j,n ∂ j and it remains to show that C ∞ ( R n ) b n is a core for ∂ j γ j,n .Let ψ ∈ dom( ∂ j γ j,n ). Then there exists a sequence { ψ k } k ∈ N in C ∞ ( R n ) b n suchthat ψ k → γ j,n ψ as k → ∞ in D ∂ j , the domain of ∂ j endowed with the graphnorm. Defining φ k := γ − j,n ψ k = γ j,n ψ k ∈ C ∞ ( R n ) b n , k ∈ N , one sees that φ k → ψ in L ( R n ) b n and ∂ j γ j,n φ k = ∂ j ψ k → ∂ j γ j,n ψ in L ( R n ) b n as k → ∞ . Thus,( γ j,n ∂ j ) ∗ = − ∂ j γ ∗ j,n = − ∂ j γ j,n = − γ j,n ∂ j .Assertion ( iii ) is a bit more involved. We shall prove it in the next two steps. (cid:3) We recall the following well-known fact in the theory of normal operators.
Theorem 6.5 (see, e.g., [52, p. 347]) . Let H be a complex separable Hilbert spaceand let A and B be self-adjoint, resolvent commuting operators acting on H . Then A + iB is closed, densely defined, and ( A + iB ) ∗ = A − iB. At this point we are ready to conclude the proof of Theorem 6.4:
Lemma 6.6.
Let H be a complex, separable Hilbert space, A , . . . , A n be resol-vent commuting skew-self-adjoint operators in H . Let { γ k } k ∈{ ,...,n } be a family ofbounded linear self-adjoint operators in H , all commuting with A j , j ∈ { , . . . , n } ,in the sense that γ k A j = A j γ k , j, k ∈ { , . . . , n } . Assume that the following equa-tion holds, γ k γ k ′ + γ k ′ γ k = 2 δ kk ′ , k, k ′ ∈ { , . . . , n } . Then P nk =1 γ k A k is closed on its natural domain T nk =1 dom( A k ) , and n X k =1 γ k A k ! ∗ = − n X k =1 γ k A k . (6.9) Proof.
We prove (6.9) by induction on n . The case n = 1 follows from ( γ A ) ∗ = A ∗ γ ∗ = − A γ = − γ A .Next, assume the assertion holds for n ∈ N and consider the sum A := n +1 X k =1 γ k A k = γ A + n +1 X k =2 γ k A k , with its natural domain T nk =1 dom( A k ). Since γ = I H , γ defines an isomorphismfrom H into itself. Hence, A is closed if and only if γ A is closed. One notes, γ n +1 X k =2 γ k A k ! ∗ = − n +1 X k =2 γ k A k γ = − n +1 X k =2 γ k γ A k = n +1 X k =2 γ γ k A k = γ n +1 X k =2 γ k A k , (6.10)in addition, γ γ A = A is skew-self-adjoint. With the help of Theorem 6.5 itremains to check whether the resolvents of A and γ P n +1 k =2 γ k A k commute. Oneobserves that for z ∈ ̺ ( A ),( z − A ) − γ n +1 X k =2 γ k A k = γ ( z − A ) − n +1 X k =2 γ k A k = γ n +1 X k =2 γ k ( z − A ) − A k ⊆ γ n +1 X k =2 γ k A k ( z − A ) − . (6.11)Adding − z ′ ( z − A ) − for some z ′ ∈ ̺ (cid:16) γ P n +1 k =2 γ k A k (cid:17) to both sides of inclusion(6.11), one obtains − z ′ ( z − A ) − + ( z − A ) − γ n +1 X k =2 γ k A k HE CALLIAS INDEX FORMULA REVISITED 45 ⊆ − z ′ ( z − A ) − + γ n +1 X k =2 γ k A k ( z − A ) − . Thus, ( z − A ) − z ′ − γ n +1 X k =2 γ k A k ! ⊆ z ′ − γ n +1 X k =2 γ k A k ! ( z − A ) − , implying z ′ − γ n +1 X k =2 γ k A k ! − ( z − A ) − ⊆ ( z − A ) − z ′ − γ n +1 X k =2 γ k A k ! − , proving assertion (6.9) since the domain of the operator on the left-hand side is allof H . (cid:3) For proving the Fredholm property of L = Q + Φ, we will employ stability of theFredholm property under relatively compact perturbations, or, in other words, thatthe essential spectrum is invariant under additive relatively compact perturbations.Thus, we need a compactness criterion and hence we recall the following compact-ness result for multiplication operators, a consequence of the Rellich–Kondrachovtheorem, see [2, Theorem 6.3] (cf. and (5.1) for the definition of H ( R n )). Theorem 6.7.
Let n ∈ N and φ ∈ L ∞ ( R n ) such that for all ε > there exists Λ > such that for all x ∈ R n \ B (0 , Λ) , | φ ( x ) | ε . Then φ : ( H ( R n ) → L ( R n ) ,f φ ( · ) f ( · ) , is compact.Proof. As H ( R n ) is a Hilbert space, it suffices to prove that weakly convergentsequences are mapped to norm-convergent sequences: Suppose that { f k } k ∈ N weaklyconverges to some f in H ( R n ) and denote M := sup k ∈ N k f k k H ( R n ) , which is finiteby the uniform boundedness principle. In particular, { f k } k ∈ N converges weaklyin H ( B (0 , Λ)) for every Λ >
0. Hence, by the Rellich–Kondrachov theorem, forall Λ > { f k } k ∈ N converges in L ( B (0 , Λ)). Next, let ε >
0. As f ∈ L ( R n ), there exists Λ > k f χ B (0 , Λ ) − f k L ε , where we denotedby χ B (0 , Λ ) the cut-off function being 1 on the ball B (0 , Λ ) and 0 elswhere. Onecan find Λ > Λ such that | φ ( x ) | ε for | x | > Λ, and k ∈ N such that for all k > k , one has k f k χ B (0 , Λ) − f χ B (0 , Λ) k L ε. Thus, for k > k one arrives at k φf k − φf k = (cid:13)(cid:13) φf k χ B (0 , Λ) − φf χ B (0 , Λ) (cid:13)(cid:13) L + (cid:13)(cid:13) φf k χ R n \ B (0 , Λ) − φf χ R n \ B (0 , Λ) (cid:13)(cid:13) L k φ k L ∞ ε + ε (2 M ) . (6.12) (cid:3) Remark . The latter theorem has the following easy but important corollary:In the situation of Theorem 6.7, let H be a Hilbert space continuously embeddedinto H ( R n ), for instance, H = H ( R n ) (cf. (5.1)), then the operator φ H→ L ofmultiplying by φ considered from H to L ( R n ) is compact. Denoting by ι : H → H ( R n ) the continuous embedding, which exists by hypothesis, one observes that φ H→ L ( R n ) = φ H ( R n ) → L ( R n ) ◦ ι, with φ H ( R n ) → L ( R n ) being the operator discussed in Theorem 6.7. Hence, theoperator φ H→ L ( R n ) is compact as a composition of a continuous and a compactoperator. ⋄ The proof of Theorem 6.3 will rest on the observation that L is Fredholm if andonly if the essential spectra of L ∗ L and LL ∗ have strictly positive lower bounds.Thus, we formulate two propositions describing the opertors L ∗ L and LL ∗ in bitmore detail: Proposition 6.9.
The operator L given by (6.2) is closed and densely defined in L ( R n ) b n d and L ∗ = −Q + Φ , dom ( L ∗ ) = dom ( L ) = H ( R n ) b n d . (6.13) Proof.
Since the operator of multiplication with the function Φ is bounded andself-adjoint, the assertion is immediate from Theorem 6.4. (cid:3)
Proposition 6.10.
Assume that Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is pointwise self-adjoint, thatis, Φ( · ) = Φ( · ) ∗ . For L = Q + Φ given by (6.2) , one then has ( cf. (6.4)) , L ∗ L = − ∆ I b n d − C + Φ and LL ∗ = − ∆ I b n d + C + Φ , (6.14) where C = n X j =1 γ j,n ( ∂ j Φ) = ( Q Φ) , (6.15) see also Remark . Moreover, dom( L ∗ L ) = dom( LL ∗ ) = H ( R n ) b n d (6.16)( see (5.1) for a definition of the latter ) .Proof. At first one observes that if ψ ∈ H k ( R n ) b n d and Lψ ∈ H k ( R n ) b n d , then ψ ∈ H k +1 ( R n ) b n d . Indeed, from Lψ = Q ψ +Φ ψ , one infers ψ + Q ψ = ψ + Lψ +Φ ψ ∈ H k ( R n ) b n d by the differentiablity of Φ. By Theorem 6.4, the operator Q is skew-self-adjoint and therefore − ∈ ̺ ( Q ). Hence, ψ = ( Q + I ) − ( Q + I ) ψ ∈ H k +1 ( R n ) b n d .Therefore, if ψ ∈ dom( L ) = H ( R n ) b n d with Lψ ∈ dom( L ∗ ) = H ( R n ) b n d , then ψ ∈ H ( R n ) b n d . On the other hand, if ψ ∈ H ( R n ) b n d , then also ψ ∈ dom( L ∗ L ).The same reasoning applies to LL ∗ .Next, we compute L ∗ L . With Proposition 6.9 one obtains L ∗ L = ( −Q + Φ)( Q + Φ) = −QQ + Φ Q − Q
Φ + Φ and LL ∗ = ( Q + Φ)( −Q + Φ) = −QQ − Φ Q + Q Φ + Φ . Recalling −QQ = − ∆ I b n d from (4.3), one concludes the proof with the observationΦ Q − Q
Φ = Φ
Q − Φ Q + C = C , applying the product rule. (cid:3) We may now come to the proof of the Fredholm property of L = Q + Φ withsmooth potential Φ satisfying for some c > | Φ( x ) | > cI d , x ∈ R n , as well assatisfying C ( x ) = ( Q Φ)( x ) → | x | → ∞ : HE CALLIAS INDEX FORMULA REVISITED 47
Proof of Theorem . By hypothesis, (Φ( x )) = | Φ( x ) | > c I d , x ∈ R n . From − ∆ I d + Φ > c I d , one deduces that the spectrum of − ∆ I d + Φ is contained in [ c , ∞ ). In particular,one concludes that the essential spectrum σ ess ( − ∆ I d + Φ ) of − ∆ I d + Φ is alsocontained in [ c , ∞ ). Since x C ( x ) = ( Q Φ) ( x ) satisfies the condition imposed onΦ in Theorem 6.7, one infers that C is − ∆ I d + Φ -compact, since the domain of thelatter (closed) operator coincides with H ( R n ) d , which is continuously embeddedinto H ( R n ) d . Recalling Proposition 6.10, that is, L ∗ L = − ∆ I b n d − C + Φ , one obtains σ ess ( L ∗ L ) = σ ess ( − ∆ I d + Φ ) ⊆ [ c , ∞ ), as the essential spectrum isinvariant under additive relatively compact perturbations (see, e.g., [71, Theorem5.35]). In particular, 0 / ∈ σ ess ( L ∗ L ) implying that L ∗ L is Fredholm. By a similarargument applied to LL ∗ , one deduces the Fredholm property of L (using thatker( L ) = ker( L ∗ L ) and ker( L ∗ ) = ker( LL ∗ )). (cid:3) In the following sections, we are interested in a particular subclass of potentialsΦ. In particular, we focus on potentials for which we may apply Theorem 3.4. Afirst main focus is set on potentials satisfying the properties stated in Definition6.11, the so-called admissible potentials. The reader is referred to Section 10 andbeyond for possible generalizations. It should be noted, however, that for moregeneral potentials the derivations and arguments are more involved than for theones mentioned in Definition 6.11. In fact, the main reason being assumption ( ii )on the invertibility of Φ everywhere. It is known (see the end of Section 10) thatthe operator L = Q + Φ has index 0 for Φ satisfying Definition 6.11. Later on, weshall see that the study of potentials being invertible on complements of large ballsaround 0 can be reduced to the study of potentials being invertible everywhereexcept on a sufficiently small ball around 0. The arguments for the latter case,in turn, rest on the perturbation theory for the Helmholtz equation, see Section11. Hence, the derivation for the index formula for potentials being invertibleeverywhere except on a sufficiently small ball can be regarded as a perturbed versionof the arguments given for admissible potentials. Therefore, we chose to presentthe core arguments for the by far simpler case of admissible potentials first.The precise notion of what we call admissible potentials reads as follows. Definition 6.11.
Let
Φ : R n → C d × d for some d, n ∈ N . We call Φ admissible , ifthe following conditions ( i ) – ( iii ) hold: ( i ) ( smoothness ) Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) . ( ii ) ( invertibility and self-adjointness ) for all x ∈ R n , Φ( x ) ∗ = Φ( x ) = Φ( x ) − . ( iii ) ( asymptotics of the derivatives ) for all α ∈ N n , there exists κ > and ε > / such that k ( ∂ α Φ)( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | > , x ∈ R n , where we employed multi-index notation and used the convention | α | = P nj =1 α j .Remark . If Ψ ∈ C ∞ b (cid:0) R n \ B (0 , C d × d (cid:1) is homogeneous of order
0, that is, forall x ∈ R n \{ } , Ψ( x ) = Ψ( x/ | x | ), then Ψ satisfies Definition 6.11 ( iii ). Indeed, one computes for x = { x j } j ∈{ ,...,n } ∈ R n \{ } and j ∈ { , . . . , n } , ∂ j (cid:18) ·|·| (cid:19) ( x ) = 1 | x | − x j | x | x ... x j ... x n , and ( ∂ j Ψ) ( x ) = ∂ j (cid:18) Ψ ◦ (cid:18) ·|·| (cid:19)(cid:19) ( x )= (cid:0) ( ∂ Ψ) ( x/ | x | ) · · · ( ∂ j Ψ) ( x/ | x | ) · · · ( ∂ n Ψ) ( x/ | x | ) (cid:1) × | x | − x j | x | x ... x j ... x n = 1 | x | n X k =1 ( ∂ k Ψ) (cid:18) x | x | (cid:19) δ kj − x k x j | x | ! , establishing the assertion. We note that Callias [22] assumes that the potential“approaches a homogeneous function of order 0 as | x | → ∞ ” such that Definition6.11 ( iii ) is satisfied. ⋄ HE CALLIAS INDEX FORMULA REVISITED 49 Derivation of the Trace Formula – The Trace Class Result
In this section, we shall prove the applicability of Theorem 3.4 for the operator L = Q + Φ (7.1)in L ( R n ) b n d as introduced in (6.2) with Q = n X j =1 γ j,n ∂ j given by (6.3) (or (4.1)) and an admissible potential Φ, see Definition 6.11. Moreprecisely, we seek to establish that the operator χ Λ B L ( z ) = zχ Λ tr b n d (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) , z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , (7.2)belongs to the trace class B (cid:0) L ( R n ) (cid:1) , where tr b n d is given in (3.1) and χ Λ is themultiplication operator of multiplying with the characteristic function of the ballcentered at 0 with radius Λ >
0, that is, χ Λ ( x ) := ( , x ∈ B (0 , Λ) , , x ∈ R n \ B (0 , Λ) . (7.3)Regarding Theorem 3.4 (with T Λ = χ Λ and S ∗ Λ = I L ( R n ) ), we are then inter-ested in computing the limit for Λ → ∞ of tr L ( R n ) ( χ Λ B L ( z )). This requiresshowing that χ Λ B L ( z ) is indeed trace class for all Λ >
0. The limit z → Λ →∞ tr L ( R n ) ( χ Λ B L ( z )) (provided it exists in an appropriate way, see (3.5) inTheorem 3.4) then corresponds to the index of L . It turns out that to computethe limit of z → Λ →∞ tr L ( R n ) ( χ Λ B L ( z )) is rather straight-forward (see also Theorem 10.1), once the respective formula is established . Themain theorem, which we shall prove in the next two sections, reads as follows. Theorem 7.1.
Let z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) with Re( z ) > − and n ∈ N > odd. Suppose that Φ is admissible ( see Definition .Then the operator χ Λ B L ( z ) with B L ( z ) and χ Λ given by (7.2) and (7.3) , respectively, is trace class, the limit f ( z ) := lim Λ →∞ tr( χ Λ B L ( z )) exists and is given by f ( z ) = (1 + z ) − n/ (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr(Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x )) x j d n − σ ( x ) , (7.4) where ε ji ...i n − denotes the ε -symbol as in Proposition A.8 . In order to deduce the latter theorem, we shall have a deeper look into the innerstructure of B L ( z ). A first step toward our goal is the following result. From now on, we shall only furnish the internal trace, introduced in Definition 3.1, of operatorsliving on an orthogonal sum of a Hilbert space, with an additional subscript. The operator trwithout subscript will always refer to the trace of a trace class operator acting in some fixedunderlying Hilbert space. In particular, for A ∈ C d × d , the expression tr( A ) denotes the sum ofthe diagonal entries. Lemma 7.2.
Let L and B L ( z ) be given by (7.1) and (7.2) , respectively. Then forall z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , B L ( z ) = tr b n d (cid:0)(cid:2) L, L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) L ∗ , L ( L ∗ L + z ) − (cid:3)(cid:1) ( where [ · , · ] represents the commutator symbol, cf. (2.2)) .Proof. Let z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ). One computes[ L, L ∗ ( LL ∗ + z ) − ] = LL ∗ ( LL ∗ + z ) − − L ∗ ( LL ∗ + z ) − L = ( LL ∗ + z ) ( LL ∗ + z ) − − z ( LL ∗ + z ) − − ( L ∗ L + z ) − L ∗ L = 1 − z ( LL ∗ + z ) − − ( L ∗ L + z ) − ( L ∗ L + z ) + ( L ∗ L + z ) − z = 1 − z ( LL ∗ + z ) − − L ∗ L + z ) − z = z ( L ∗ L + z ) − − z ( LL ∗ + z ) − , and, interchanging the roles of L and L ∗ , one concludes[ L ∗ , L ( L ∗ L + z ) − ] = z ( LL ∗ + z ) − − z ( L ∗ L + z ) − . (cid:3) The forthcoming Proposition 7.4 gives a more detailed description of the com-mutators describing B L ( z ) just derived in Lemma 7.2. First, we need a prerequisitof a more general nature. Lemma 7.3.
Let n ∈ N , B ∈ B (cid:0) L ( R n ) b n d , L ( R n ) b n d (cid:1) and let Q and γ j,n , j ∈{ , . . . , n } , as in (6.3) and in Remark , respectively. Then, on the commonnatural domain of the operator sums involved, one has tr b n d ([ Q , B ]) = n X j =1 tr b n d ([ ∂ j , γ j,n B ]) = n X j =1 tr b n d ([ ∂ j , Bγ j,n ]) . Proof.
One computes with the help of Proposition 3.3 and the fact γ j,n ∂ j = ∂ j γ j,n ,tr b n d ( Q B − B Q ) = n X j =1 tr b n d ( γ j,n ∂ j B − Bγ j,n ∂ j )= n X j =1 [tr b n d ( γ j,n ∂ j B ) − tr b n d (( Bγ j,n ) ∂ j )]= n X j =1 [tr b n d ( ∂ j γ j,n B ) − tr b n d (( γ j,n B ) ∂ j )]= n X j =1 tr b n d ([ ∂ j , γ j,n B ]) . The second equality can be shown similarly. (cid:3)
The following proposition represents the core of the derivation of the index for-mula. Once it is proven that χ Λ B L ( z ) is trace class, with the trace being computedas the integral over the diagonal of the respective integral kernel, equation (7.5) willbe the key for computing the trace. More precisely, the first summand is a sum ofcommutators of certain operators with partial derivatives. For the respective inte-gral kernels, this will give us an expression as in Lemma 5.6 (see also (5.5)), whichwill enable us to use Gauss’ divergence theorem, explaining the surface integral in HE CALLIAS INDEX FORMULA REVISITED 51 (7.4). Furthermore, the second summand in equation (7.5) as can be seen in equa-tion (7.7) is basically a commutator of an integral operator and a multiplicationoperator. The integral kernels of this type of operators have been shown to vanishon the diagonal in Proposition 5.5, thus, (7.7) will give a vanishing contribution tothe trace of B L ( z ). Proposition 7.4 ([22, Proposition 1, p. 219]) . Let L be given by (7.1) and z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) . Then B L ( z ) given by (7.2) satisfies B L ( z ) = n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) + A L ( z ) , (7.5) where J jL ( z ) = tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) + tr b n d (cid:0) L ∗ ( LL ∗ + z ) − γ j,n (cid:1) , j ∈ { , . . . , n } , (7.6) and A L ( z ) = tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , L ( L ∗ L + z ) − (cid:3)(cid:1) , (7.7) with γ j,n as in Remark or Appendix A .Proof. One recalls that L ∗ = −Q + Φ from Proposition 6.9. From Lemma 7.2, oneinfers that2 B L ( z ) = tr b n d (cid:0)(cid:2) L, L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) L ∗ , L ( L ∗ L + z ) − (cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Q + Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) − Q + Φ , L ( L ∗ L + z ) − (cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Q , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) + tr b n d (cid:0)(cid:2) Q , L ( L ∗ L + z ) − (cid:3)(cid:1) + tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , L ( L ∗ L + z ) − (cid:3)(cid:1) . The equationstr b n d (cid:0)(cid:2) Q , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) = n X j =1 tr b n d (cid:0)(cid:2) ∂ j , L ∗ ( LL ∗ + z ) − γ j,n (cid:3)(cid:1) , and tr b n d (cid:0)(cid:2) Q , L ( L ∗ L + z ) − (cid:3)(cid:1) = n X j =1 tr b n d (cid:0)(cid:2) ∂ j , L ( L ∗ L + z ) − γ j,n (cid:3)(cid:1) follow from Lemma 7.3. (cid:3) Next, we show that (a modification in the sense of Theorem 3.4 of) B L ( z ) givesrise to trace class operators. Before doing so in Theorem 7.8, we need a differentrepresentation of B L ( z ) in terms of powers of the resolvent of the (free) Laplacian.One notes that for z ∈ C , with Re( z ) > sup x ∈ R n max j k ∂ j Φ( x ) k −
1, one has k CR z k <
1, with C given by (6.15). Hence, by Proposition 6.10, equation (6.14),one obtains ( L ∗ L + z ) − = ( − ∆ I b n d − C + (1 + z )) − = (( − ∆ I b n d + (1 + z )) ( I b n d − R z C )) − = ( I b n d − R z C ) − R z = ∞ X k =0 ( R z C ) k R z , , (7.8) and, similarly,( LL ∗ + z ) − = ( − ∆ I b n d + C + (1 + z )) − = ∞ X k =0 ( − R z C ) k R z . (7.9)Consequently, by analytic continuation, one obtains for z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ )with Re( z ) > − L ∗ L + z ) − = N X k =0 ( R z C ) k R z + ( R z C ) N +1 ( L ∗ L + z ) − , (7.10)and ( LL ∗ + z ) − = N X k =0 ( − R z C ) k R z + ( − R z C ) N +1 ( LL ∗ + z ) − , (7.11)for all N ∈ N . Focussing on resolvent differences, one gets the following proposition: Proposition 7.5.
Let z ∈ C Re > − . One recalls L = Q + Φ as in (7.1) , C = ( Q Φ) from (6.15) , and R z in (4.6) . ( i ) If Re( z ) > sup x ∈ R n max j k ( ∂ j Φ)( x ) k − , then z ∈ ̺ ( − L ∗ L ) ∩ ( − LL ∗ ) and ( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 ∞ X k =0 ( R z C ) k +1 R z = 2 ∞ X k =0 R z ( CR z ) k +1 , as well as ( L ∗ L + z ) − + ( LL ∗ + z ) − = 2 ∞ X k =0 ( R z C ) k R z = 2 ∞ X k =0 R z ( CR z ) k . ( ii ) If z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) and Re( z ) > − , then for all N ∈ N , ( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 N X k =0 R z ( CR z ) k +1 + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) N +2 = 2 N X k =0 R z ( CR z ) k +1 + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) N +3 , and ( L ∗ L + z ) − + ( LL ∗ + z ) − = 2 N X k =0 R z ( CR z ) k + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) N +2 . Proof. ( i ) This is a direct consequence of equations (7.8) and (7.9).( ii ) For z as in part ( i ) one computes, similarly to (7.10) and (7.11), with the helpof item ( i ) for N ∈ N ,( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 N X k =0 R z ( CR z ) k +1 + 2 ∞ X k = N +1 R z ( CR z ) k +1HE CALLIAS INDEX FORMULA REVISITED 53 = 2 N X k =0 R z ( CR z ) k +1 + 2 ∞ X k =0 R z ( CR z ) k +2 N +2+1 = 2 N X k =0 R z ( CR z ) k +1 + 2 ∞ X k =0 R z ( CR z ) k +1 ( CR z ) N +2 = 2 N X k =0 R z ( CR z ) k +1 + 2 ∞ X k =0 R z ( CR z ) k ( CR z ) N +3 . Hence,( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 N X k =0 R z ( CR z ) k +1 + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) N +2 = 2 N X k =0 R z ( CR z ) k +1 + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) N +3 , again by part ( i ). Analytic continuation implies the asserted equalities. (The secondterm in item ( ii ) is treated analogously). (cid:3) Before starting the proof that χ Λ B L ( z ), with B L ( z ) given by (7.2), is traceclass, and then prove the trace formula in Theorem 7.1 for this operator, a closerinspection of the operators occuring in Proposition 7.4 with the help of Proposition7.5 is in order. In particular, the principal aim of Lemma 7.7, is twofold: on onehand, we will prove that the power series representation of B L ( z ), basically derivedin Proposition 7.5, starts with an operator essentially of the form R z ( CR z ) k +1 for some k ∈ N . For this kind of operators we have a trace class criterion at hand,Theorem 4.7 together with Corollary 4.3. On the other hand, we also prove repre-sentation formulas for the operators in (7.6) and (7.7). These formulas also startwith operators involving high powers of R z . This leads to continuity and differ-entiability properties for the corresponding integral kernels enabling the applicationof Proposition 5.5 and Lemma 5.6.The key idea for proving Lemma 7.7, contained in Lemma 7.6, is to use thecancellation properties of the Euclidean Dirac algebra under the trace sign. For theEuclidean Dirac algebra we refer to Definition A.3; moreover, we refer to Proposi-tion A.8 for the cancellation properties. Lemma 7.6.
Let L = Q + Φ be given by (7.1) . Let z ∈ C with Re( z ) > − and z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) and recall C = [ Q, Φ] , k ∈ N odd. If either k < n or n iseven, then tr b n d (cid:16) R z (cid:0) CR z (cid:1) k (cid:17) = 0 . Proof.
One observes using the fact that γ j,n , j ∈ { , . . . , n } , commutes with both R z and ( ∂ ℓ Φ), ℓ ∈ { , . . . , n } (cf. Remark 6.1), that R z (cid:0) CR z (cid:1) k = R z (cid:0) n X ℓ =1 γ ℓ,n ( ∂ ℓ Φ) R z (cid:1) k = R z n X ℓ , ··· ,ℓ k =1 γ ℓ ,n ( ∂ ℓ Φ) R z · · · γ ℓ k ,n ( ∂ ℓ k Φ) R z = n X ℓ ,...,ℓ k =1 γ ℓ ,n · · · γ ℓ k ,n R z ( ∂ ℓ Φ) R z · · · ( ∂ ℓ k Φ) R z . Next, employingtr b n d (cid:16) γ ℓ ,n · · · γ ℓ k ,n R z ( ∂ ℓ Φ) R z · · · ( ∂ ℓ k Φ) R z (cid:17) = tr b n (cid:16) γ ℓ ,n · · · γ ℓ k ,n (cid:17) tr d (cid:16) R z ( ∂ ℓ Φ) R z · · · ( ∂ ℓ k Φ) R z (cid:17) for all i , . . . , i k ∈ { , . . . , n } , one concludes thattr b n d (cid:16) R z (cid:0) CR z (cid:1) k (cid:17) = 0 , by Proposition A.8. (cid:3) Lemma 7.7.
Let L = Q + Φ be given by (7.1) . Let z ∈ C with Re( z ) > − and z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) . One recalls B L ( z ) , J jL ( z ) , and A L ( z ) given by (7.2) , (7.6) , and (7.7) , respectively, as well as R z given by (4.6) . Then the followingassertions hold: ( i ) For all odd n ∈ N > , B L ( z ) = n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) + A L ( z ) (7.12)= z tr b n d (cid:0) R z C ) n R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:1) , and, for all j ∈ { , . . . , n } , J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) , and A L ( z ) = tr b n d (cid:0)(cid:2) Φ , Φ (cid:0) R z C ) n R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR z ) n +1 (cid:1)(cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , Q (cid:0) R z C ) n − R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR z ) n (cid:1)(cid:3)(cid:1) . ( ii ) For all even n ∈ N , B L ( z ) = 0 . (7.13) Proof.
From Proposition 7.5, one has for Re( z ) > − N ∈ N ,( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 N X k =0 R z ( CR z ) k +1 + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) N +2 . In addition, using Lemma 7.6, one deduces that for n even,tr b n d (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) = 0 , HE CALLIAS INDEX FORMULA REVISITED 55 and, for n odd,tr b n d (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) = tr b n d (cid:0) R z C ) n R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:1) . This proves (7.12).In a similar fashion, using again Proposition 7.5 and the “cyclicity property” oftr b n d (see Proposition 3.3), one obtainstr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) + tr b n d (cid:0) L ∗ ( LL ∗ + z ) − γ j,n (cid:1) = tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n + L ∗ ( LL ∗ + z ) − γ j,n (cid:1) = tr b n d (cid:0) ( Q + Φ) ( L ∗ L + z ) − γ j,n + ( −Q + Φ) ( LL ∗ + z ) − γ j,n (cid:1) = tr b n d (cid:0) Q ( L ∗ L + z ) − γ j,n − Q ( LL ∗ + z ) − γ j,n (cid:1) + tr b n d (cid:0) Φ (cid:0) ( L ∗ L + z ) − γ j,n + ( LL ∗ + z ) − γ j,n (cid:1)(cid:1) = tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1)(cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1)(cid:1) = 2 tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) , and A L ( z ) = tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , L ( L ∗ L + z ) − (cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − − L ( L ∗ L + z ) − (cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Φ , Φ ( LL ∗ + z ) − − Φ ( L ∗ L + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , Q ( L ∗ L + z ) − + Q ( LL ∗ + z ) − (cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Φ , Φ (cid:0) ( LL ∗ + z ) − − ( L ∗ L + z ) − (cid:1)(cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1)(cid:3)(cid:1) = tr b n d (cid:0)(cid:2) Φ , Φ (cid:0) R z C ) n R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR z ) n +1 (cid:1)(cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , Q (cid:0) R z C ) n − R z + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) × ( CR z ) n (cid:1)(cid:3)(cid:1) . (cid:3) One important upshot of Lemma 7.7 is the fact (7.13), implying that only odddimensions are of interest when computing the index of L . Thus, we will focus onthe case n odd, only.The next theorem concludes this section and asserts that the trace class assump-tions on B L ( z ) in Theorem 3.4 are satisfied for B L ( z ) given by (7.2). As the se-quence { T Λ } Λ we shall use { χ Λ } Λ the sequence of multiplication operators inducedby multiplying with the cut-off (characteristic) function χ Λ . The sequence { S ∗ Λ } Λ is set to be the constant sequence S Λ = I L ( R n ) for all Λ. Clearly, χ Λ ∈ L n +1 ( R n )for all Λ > Theorem 7.8.
Let n ∈ N > odd, L = Q +Φ given by (7.1) . Then there exists δ > such that for all z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) and Λ > , the operator χ Λ B L ( z ) with B L ( z ) given by (7.2) is trace class with z tr( | χ Λ B L ( z ) | ) bounded on B (0 , δ ) \{ } .Proof. We start by showing that z χ Λ R z (cid:0) ( CR z ) n (cid:1) is trace class withtrace class norm being bounded around a neighborhood of 0, where C = ( Q Φ) = P nj =1 γ j,n ( ∂ j Φ) is given by (6.15), see also Remark 2.1, and R z is given by (4.6).Using n = 2 b n + 1 we write χ Λ R z (cid:0) CR z ) n (cid:1) = (cid:16) χ Λ R z ( CR z ) b n (cid:17) (cid:16) ( CR z ) b n +1 (cid:17) . By Theorem 4.7 the operators (cid:16) χ Λ R z ( CR z ) b n (cid:17) and (cid:16) ( CR z ) b n +1 (cid:17) are Hilbert–Schmidt by the admissability of Φ (in this context, see, in particular,Definition 6.11 ( iii )). Moreover, the boundedness of z χ Λ tr b n d ( R z C ) n withrespect to the norm in B (cid:0) L ( R n ) (cid:1) around a neighborhood of 0, now follows fromTheorem 4.2 together with the estimates in Theorem 4.7 and Lemma 4.5 (we notethat we apply these statements for µ = 1 + z with z ∈ C Re > − ).One recalls (employing the spectral theorem) that for all self-adjoint operators A on a Hilbert space H with 0 being an isolated eigenvalue, the operator family z z ( A + z ) − is uniformly bounded on B (0 , δ ) for some δ >
0. By Lemma 7.7,(7.12), the uniform boundedness of z z ( A + z ) − on B (0 , δ ) for some δ >
0, andthe ideal property for trace class operators, it remains to show that ( CR z ) n +1 is trace class, with trace class norm bounded for z ∈ B (0 , δ ′ ) for some sufficientlysmall δ ′ >
0. For n = 2 b n + 1, one observes that ( CR z ) n +1 is a sum of operatorsof the formΨ · · · R z Ψ n +1 R z = (Ψ · · · R z Ψ b n +1 R z ) (Ψ b n +2 · · · R z Ψ b n +2 R z ) , where Ψ j are multiplication operators with bounded C ∞ -functions with the prop-erty that for some constant κ > | Ψ j ( x ) | κ | x | , x ∈ R n . For deriving the traceclass property ofΨ · · · R z Ψ n +1 R z = (Ψ · · · R z Ψ b n +1 R z ) (Ψ b n +2 · · · R z Ψ b n +2 R z ) , we use Theorem 4.7 and Lemma 4.5. Let z ∈ ( − , i ) oneestimates for some κ ′ >
0, depending on b n , κ and z , and all z ∈ C > z , k (Ψ · · · R z Ψ b n +1 R z ) k B b n +1 Y j =1 k Ψ j R z k κ ′ b n +1 Y j =1 k Ψ j k L n +1 , where we used Lemma 4.5 in the last estimate. The same argument applies to(Ψ b n +2 · · · R z Ψ b n +2 R z ) . This concludes the proof since ( CR z ) n +1 is trace class by Theorem 4.2. (cid:3) Remark . We note that the method of proof of Theorem 7.8 shows that the traceof χ Λ B L ( z ) can be computed as the integral over the diagonal of the respectiveintegral kernel. In fact, we have shown that χ Λ B L ( z ) may be represented as sumsof products of two Hilbert–Schmidt operators leading to the trace formula given inCorollary 4.3. ⋄ HE CALLIAS INDEX FORMULA REVISITED 57 Derivation of the Trace Formula – Diagonal Estimates
In this section, we shall compute the trace of χ Λ B L ( z ), Λ > z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) ∩ C Re > − , with B L given by (3.2). After stating the next lemma (neededto be able to apply Lemma 5.6 and Proposition 5.5 to the sum in (7.5)) we willoutline the strategy of the proof.We note that for the application of Lemma 5.6 to the first summand in (7.5),one needs to establish continuous differentiability of the integral kernel of (7.6).In this context we emphasize the different regularity of the kernels of (7.6) for n = 3 and n >
5, necessitating modifications for the case n = 3 due to the lack ofdifferentiability of (7.6). Lemma 8.1 ([22, Lemma 4, p. 224]) . Let n ∈ N > odd, L = Q + Φ be givenby (7.1) , and let z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , with Re( z ) > − . Denote the integralkernels of the following operators J jL ( z ) = tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) − tr b n d (cid:0) L ∗ ( LL ∗ + z ) − γ j,n (cid:1) ,A L ( z ) = tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , L ( L ∗ L + z ) − (cid:3)(cid:1) , by G J,j,z , j ∈ { , . . . , n } , and G A,z , respectively. Then G A,z is continuous andsatisfies G A,z ( x, y ) → if y → x for all x ∈ R n . If n > , G J,j,z is continuouslydifferentiable on R n × R n . Proof.
Appealing to Lemma 7.7, one recalls with R z , Q , and C given by (4.6),(6.3), and (6.15), respectively, J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) , j ∈ { , . . . , n } , and A L ( z ) = tr b n d (cid:0)(cid:2) Φ , Φ (cid:0) R z C ) n R z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR z ) n +1 (cid:1)(cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , Q (cid:0) R z C ) n − R z + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) × ( CR z ) n (cid:1)(cid:3)(cid:1) . By Proposition 5.4 (one recalls that Φ is admissible and hence Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) by Definition 6.11 ( i )), one gets for all ℓ ∈ R , Qγ j,n ( R z C ) n − R z ∈ B (cid:0) H ℓ ( R n ) b n d , H ℓ +2( n − − ( R n ) b n d (cid:1) . For n >
5, one obtains from (2( n −
2) + 2 −
1) = n − > , the continuity of G J,j,z by Corollary 5.3. Moreover, since (2( n −
2) + 2 − − n − n − > , Corollary5.3 also implies continuous differentiability of G J,j,z . Similar arguments ensure thecontinuity of the integral kernel of A L ( z ) (for n > n >
3, theintegral kernel of A L ( z ) vanishes on the diagonal by Proposition 5.5. (cid:3) Next, we outline the idea for computing the trace of χ Λ B L ( z ). By Theorem 3.4and Theorem 7.1, we know that the limit lim Λ →∞ tr( χ Λ B L (0)) exists. However,in order to derive the explicit formula asserted in Theorem 7.1 also for z in aneighborhood of 0, some work is required. As it will turn out, for z with large real part – at least for a sequence { Λ k } k ∈ N – we can show that an expression similar tothe one in Theorem 7.1 is valid.For achieving the existence of the limit (without using sequences) for z in aneighborhood of 0, we intend to employ Montel’s theorem. One recalls that for anopen set U ⊆ C , a set G ⊆ C U := { f | f : U → C } is called locally bounded , if for allcompact Ω ⊂ U , sup f ∈G sup z ∈ Ω | f ( z ) | < ∞ . (8.1) Theorem 8.2 (Montel’s theorem, see, e.g., [35], p. 146–154) . Let U ⊆ C open, { f Λ } Λ ∈ N a locally bounded family of analytic functions on U . Then there existsa subsequence { f Λ k } k ∈ N and an analytic function g on U such that f Λ k → g as k → ∞ in the compact open topology ( i.e., for any compact set Ω ⊂ U , the sequence { f Λ k | Ω } k ∈ N converges uniformly to g | Ω ) . For our particular application of Montel’s theorem, we need to show that thefamily of analytic functions { z tr( χ Λ B L ( z )) } Λ constitutes a locally bounded family. Thus, one needs to show that for all compactΩ ⊂ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ),sup Λ > sup z ∈ Ω | tr( χ Λ B L ( z )) | < ∞ . (8.2)For this assertion, it is crucial that some integral kernels involved in the computationof the trace vanish on the diagonal, see, for instance, Proposition 5.5. We note thatgenerally, the expression sup Λ > sup z ∈ Ω tr( | ( χ Λ B L ( z )) | ) , (8.3)cannot be finite, as the example constructed in Appendix B demonstrates. In orderto prove (8.2), we actually show for all Ω ⊂ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) compact,sup Λ > sup z ∈ Ω | z tr( χ Λ B L ( z )) | < ∞ , (8.4)and then appeal to the fact that condition (8.4) together with Theorem 7.8 implies(8.2), as the next result confirms: Lemma 8.3.
Assume that { φ k } k ∈ N is a sequence of analytic ( scalar-valued ) func-tions on B C (0 , . Assume that { z zφ k ( z ) } k ∈ N is locally bounded on B (0 , .Then { φ k } k ∈ N is locally bounded on B (0 , .Proof. Assume that { φ k } k ∈ N is not locally bounded on B (0 , { φ k ℓ } ℓ ∈ N and a corresponding sequence of complex numbers { z k ℓ } ℓ ∈ N with the property that z k ℓ → | φ k ℓ ( z k ℓ ) | → ∞ as ℓ → ∞ . Since { ψ ℓ } ℓ ∈ N := { z zφ k ℓ ( z ) } ℓ ∈ N is locally bounded on B (0 ,
1) there exists an accumulation point ψ in the compactopen topology of analytic functions H ( B (0 , B (0 ,
1) by Montel’s theorem.Without loss of generality, one can assume that ψ ℓ → ψ in H ( B (0 , ℓ → ∞ .By construction, one has ψ ℓ (0) = 0 and for some r > (cid:12)(cid:12)(cid:12)(cid:12) z ψ ℓ ( z ) − ψ ′ (0) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) z ( ψ ℓ ( z ) − ψ ℓ (0)) − ψ ′ ℓ (0) (cid:12)(cid:12)(cid:12)(cid:12) + | ψ ′ ℓ (0) − ψ ′ (0) | HE CALLIAS INDEX FORMULA REVISITED 59 sup z ∈ B (0 ,r ) | ( ψ ′ ℓ ( z ) − ψ ′ ℓ (0)) | + | ψ ′ ℓ (0) − ψ ′ (0) | for all z ∈ B (0 , r ) \{ } . Since ψ ′ ℓ → ψ ′ uniformly on compacts, it follows thatlim sup ℓ →∞ sup z ∈ B (0 ,r ) \{ } (cid:12)(cid:12)(cid:12)(cid:12) z ψ ℓ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ . However, for ℓ sufficiently large, one concludessup z ∈ B (0 ,r ) \{ } (cid:12)(cid:12)(cid:12)(cid:12) z ψ ℓ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12) z k ℓ ψ ℓ ( z k ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) = | φ k ℓ ( z k ℓ ) | −→ ℓ →∞ ∞ , a contradiction. (cid:3) Remark . It turns out that the analyticity hypothesis in Lemma 8.3 is crucial.Indeed, for every n ∈ N , there exists a C ∞ -function ψ n : [0 , → [0 , ∞ ) with theproperties, ψ n | (0 , / (2 n )) = 0 , ψ n ( x ) ψ n (cid:18) n (cid:19) = n, ψ n | (2 /n, = 0 . Then ψ n (0) = 0 and 0 xψ n ( x ) (2 /n ) n = 2. Considering φ n ( x + iy ) := ψ n ( | x + iy | ) for x, y ∈ R , x + iy ∈ B (0 , n ∈ N , one gets that φ n is real differentiableand the assumptions of Lemma 8.3, except for analyticity, are all satisfied. Inaddition, φ n (0) = 0, however, φ n (1 /n ) = n → ∞ as n → ∞ . Thus, { φ n } n ∈ N is notlocally bounded on B (0 , ⋄ The next aim of this section is to establish Theorem 8.7, that is, an importantstep for obtaining (8.2). The terms to be discussed in Theorem 8.7 split up into aleading order term and the rest. The first term will be studied in Lemma 8.5 andthe second one in Lemma 8.6. The strategy of proof in these lemmas is the same.It rests on the following observation: Let U ⊆ C open, U ∋ z T ( z ) ∈ B (cid:0) L ( R n ) (cid:1) .Assume that for all z ∈ U we have T ( z ) ∈ B (cid:0) L ( R n ) (cid:1) and that z tr( | T ( z ) | ) islocally bounded. Then { z tr( χ Λ T ( z )) } Λ > is locally bounded as well. Indeed, the assertion follows from the boundednessof the family { χ Λ } Λ > as bounded linear (multiplication) operators in B ( L ( R n ))and the ideal property of the trace class. In the situations to be considered inthe following, the trace class property for T ( z ) will be shown with the help of theresults of Section 4. Lemma 8.5.
Let L = Q + Φ be given by (7.1) and for z ∈ C with Re( z ) > − let R z be given by (4.6) and C as in (6.15) , n ∈ N > odd. For j ∈ { , . . . , n } ,let γ j,n ∈ C b n × b n as in Remark and χ Λ as in (7.3) , Λ > . For z ∈ C Re > − consider ψ Λ ( z ) := χ Λ tr b n d (cid:0) [ Q , Φ ( CR z ) n ] (cid:1) and e ψ Λ ( z ) := χ Λ tr b n d (cid:0) [ Q , Q ( CR z ) n ] (cid:1) . Then for all z ∈ C Re > − , the operators ψ Λ ( z ) , e ψ Λ ( z ) are trace class and the families { z tr L ( R n ) ( ψ Λ ( z )) } Λ > and (cid:8) z tr L ( R n ) (cid:0) e ψ Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) . Proof.
First we deal with ψ Λ ( z ). One computes, ψ Λ ( z ) = χ Λ tr b n d (cid:0) [ Q , Φ ( CR z ) n ] (cid:1) = χ Λ tr b n d (cid:0) Q Φ ( CR z ) n − Φ ( CR z ) n Q (cid:1) . Before we discuss the latter operator, we note that Q Φ ( CR z ) n = Φ Q ( CR z ) n + [ Q , Φ] ( CR z ) n = Φ (cid:18) n X j =1 ( CR z ) j − [ Q , C ] R z ( CR z ) n − j + ( CR z ) n Q (cid:19) + [ Q , Φ] ( CR z ) n , where the latter equality follows via an induction argument. Hence, ψ Λ ( z ) = χ Λ tr b n d (cid:18) Φ (cid:18) n X j =1 ( CR z ) j − [ Q , C ] R z ( CR z ) n − j (cid:19) + [ Q , Φ]( CR z ) n (cid:19) . (8.5)Next, with the results of Section 4, we will deduce that the operator family z (cid:18) Φ (cid:18) n X j =1 ( CR z ) j − [ Q , C ] R z ( CR z ) n − j (cid:19) + [ Q , Φ] ( CR z ) n (cid:19) (8.6)is trace class, which – together with the estimates in Lemma 4.5 – establishes theassertion for ψ Λ : Indeed, the only difference between (8.5) and (8.6) is the prefactor χ Λ . So we get the assertion with the help of the reasoning prior to Lemma 8.5. Inorder to observe that each summand in (8.6) is trace class, we proceed as follows.Recall n = 2 b n + 1 and let j ∈ { , . . . , b n } (the case n − j ∈ { , . . . , b n } can be dealtwith similarly). Then, by the admissability of Φ (see Hypothesis 6.11), one infersthat [ Q , C ] is a multiplication operator with | [ Q , C ]( x ) | κ (1 + | x | ) − − ε , x ∈ R n . Hence, as 1 + ε > / CR z ) j − [ Q , C ] R z ( CR z ) b n − j is Hilbert–Schmidt. Using Theorem 4.7, one deduces that ( CR z ) b n +1 is alsoHilbert–Schmidt and thus( CR z ) j − [ Q , C ] R z ( CR z ) b n − j ( CR z ) b n +1 is trace class, by Theorem 4.2.For e ψ Λ one proceeds similarly. First one notes that e ψ Λ ( z ) = χ Λ tr b n d (cid:18) Q (cid:18) n X j =1 ( CR z ) j − [ Q , C ] R z ( CR z ) n − j (cid:19)(cid:19) . (8.7)Applying Theorems 4.7, 4.9, and 4.2, one infers the assertion for e ψ Λ . However,one has to use the respective assertions, where some of the resolvents of the Lapla-cian is replaced by Q times the resolvents. Indeed, in the sum in (8.7), the termfor j = 1 yields Q [ Q , C ] R z ( CR z ) n − = [ Q , [ Q , C ]] R z ( CR z ) n − + [ Q , C ] Q R z ( CR z ) n − = [ Q , [ Q , C ]] R z ( CR z ) n − + [ Q , C ] R z [ Q , C ] R z ( CR z ) n − + [ Q , C ] R z C Q R z ( CR z ) n − , and for j ′ ∈ { , . . . , n } one obtains Q CR z ( CR z ) j ′ − [ Q , C ] R z ( CR z ) n − j ′ = [ Q , C ] R z ( CR z ) j ′ − [ Q , C ] R z ( CR z ) n − j ′ + C Q R z ( CR z ) j ′ − [ Q , C ] R z ( CR z ) n − j ′ . (cid:3) The next lemma is the reason, why we have to invoke Lemma 8.3 in our argu-ment. The crucial point is that we can use the Neumann series expressions for theresolvents ( L ∗ L + z ) − and ( LL ∗ + z ) − only for z with large real part. But for z in the vicinity of 0, we do not have such a representation. Using again the idealproperty for trace class operators, we can, however, bound z ( L ∗ L + z ) − for small z in the B ( L ( R n ))-norm. Introducing the sector Σ z ,ϑ ⊂ C byΣ z ,ϑ := { z ∈ C | Re( z ) > z , | arg( µ ) | < ϑ } , (8.8)for some z ∈ R and ϑ ∈ (0 , π/ Lemma 8.6.
Let L = Q + Φ be given by (7.1) and for z ∈ C with Re( z ) > − , let R z be given by (4.6) and C as in (6.15) , χ Λ as in (7.3) , Λ > . For j ∈ { , . . . , n } ,let γ j,n ∈ C b n × b n as in Remark . For z ∈ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) consider η Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:3)(cid:1) and e η Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , (cid:0) Q (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:1)(cid:3)(cid:1) . Then for all z ∈ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) , the operators η Λ ( z ) , e η Λ ( z ) are traceclass. There exists δ ∈ ( − , , ϑ ∈ (0 , π/ such that the families { Σ δ,ϑ ∪ C Re > ∋ z z tr L ( R n ) ( η Λ ( z )) } Λ > and (cid:8) Σ δ,ϑ ∪ C Re > ∋ z z tr L ( R n ) (cid:0)e η Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) .Proof. By the Fredholm property of L there exist δ ∈ ( − ,
0) and ϑ ∈ (0 , π/
2) suchthat Σ δ,ϑ \{ } ∋ z z ( L ∗ L + z ) − and Σ δ,ϑ \{ } ∋ z z ( LL ∗ + z ) − have analytic extensions to Σ δ,ϑ . Let Ω ⊂ Σ δ,ϑ ∪ C Re > be compact. One notesthat Ω ∋ z z ( L ∗ L + z ) − and Ω ∋ z z ( LL ∗ + z ) − define bounded families of bounded linear operators from L ( R n ) b n d to H ( R n ) b n d .Indeed, by Proposition 6.10, one infers that φ
7→ k ( L ∗ L +1) φ k and φ
7→ k ( LL ∗ +1) φ k define equivalent norms on H ( R n ) b n d . Hence, for φ ∈ L ( R n ) b n d and z ∈ Ω \{ } one computes (cid:13)(cid:13) ( L ∗ L + 1) z ( L ∗ L + z ) − φ (cid:13)(cid:13) = | z | (cid:13)(cid:13) ( L ∗ L + z + 1 − z )( L ∗ L + z ) − φ (cid:13)(cid:13) | z |k φ k + | z || (1 − z ) | | z | k φ k . Next, consider η Λ ( z ) = χ Λ tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:3)(cid:1) = χ Λ tr b n d (cid:0) Q Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 − Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 Q (cid:1) . For the first summand one observes that Q Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 = C (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 + Φ Q (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 . Employing our observation at the beginning of the proof and Theorem 6.4, onerealizes that Ω ∋ z
7→ Q z (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) defines a bounded family of bounded linear operators in L ( R n ) b n d . Thus, sinceΩ ∋ z ( CR z ) n +1 is a family of trace class operators,Ω ∋ z zχ Λ tr b n d (cid:0) Q Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:1) = tr b n d (cid:0) zχ Λ (cid:0) Q Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:1)(cid:1) is uniformly bounded in B , with bound independently of Λ >
0, upon appealingto the ideal property of trace class operators.The second summand requires the observation that( CR z ) n +1 Q = ( CR z ) n ( CR z ) Q = ( CR z ) n ( C Q R z )defines a bounded family of trace class operators for z ∈ Ω, proving the assertionfor η Λ .The corresponding assertion for e η Λ is conceptually the same. In fact, it followsfrom the observation thatΩ ∋ z z (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) = ∆ I b n d z (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) is a bounded family of bounded linear operators by our preliminary observationthat Ω ∋ z → z ( L ∗ L + z ) − and Ω ∋ z → z ( LL ∗ + z ) − define uniformly boundedoperator families from L ( R n ) b n d to H ( R n ) b n d , as well as using again the fact thatΩ ∋ z ( CR z ) n +1 Q and Ω ∋ z ( CR z ) n +1 constitute bounded families of trace class operators. (cid:3) Lemmas 8.5 and 8.6 can be summarized as follows.
Theorem 8.7.
Let n ∈ N > odd, let L = Q + Φ be given by (7.1) and for z ∈ C with Re( z ) > − let R z be given by (4.6) and C as in (6.15) . For j ∈ { , . . . , n } ,let γ j,n ∈ C b n × b n as in Remark . For z ∈ C Re > − ∩ ̺ ( − LL ∗ ) ∩ ̺ ( L ∗ L ) , introduce φ Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) and e φ Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , (cid:0) Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1)(cid:3)(cid:1) . HE CALLIAS INDEX FORMULA REVISITED 63
Then for all z ∈ C Re > − ∩ ̺ ( − LL ∗ ) ∩ ̺ ( L ∗ L ) , the operators φ Λ ( z ) , e φ Λ ( z ) are traceclass. There exists δ ∈ ( − , , ϑ ∈ (0 , π/ such that the families { Σ δ,ϑ ∪ C Re > ∋ z tr L ( R n ) ( zφ Λ ( z )) } Λ > and (cid:8) Σ δ,ϑ ∪ C Re > ∋ z tr L ( R n ) (cid:0) z e φ Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) .Proof. One recalls from equations (7.8) and (7.9) the expressions( L ∗ L + z ) − = I + ( L ∗ L + z ) − CR z , ( LL ∗ + z ) − = I − ( LL ∗ + z ) − CR z . Hence, one gets φ Λ ( z ) = χ Λ tr b n d (cid:0) Q , Φ ( CR z ) n ] (cid:1) + χ Λ tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:3)(cid:1) = 2 ψ Λ ( z ) + η Λ ( z ) , and e φ Λ ( z ) = χ Λ tr b n d (cid:0) Q , Q ( CR z ) n ] (cid:1) + χ Λ tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR z ) n +1 (cid:3)(cid:1) = 2 e ψ Λ ( z ) + e η Λ ( z ) , with the functions introduced in Lemmas 8.5 and 8.6. Thus, the assertion on thelocal boundedness follows from these two lemmas. (cid:3) The forthcoming statements are used for showing that for computing the tracethe only term that matters is discussed in Proposition 8.13. We recall that byRemark 7.9, one can compute the trace of χ Λ B L ( z ) as the integral over the diagonalof its integral kernel. So the estimates on the diagonal derived in Section 5 will beused in the following. We shall elaborate on this idea further after having statedthe next two auxiliaury results. Both these results serve to show that some integralkernels actually vanish on the diagonal. Lemma 8.8.
Let n ∈ N > be odd, z ∈ C with Re( z ) > − . Let R z , Q , C , and γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , be given by (4.6) , (6.3) , (6.15) and as in Remark , respectively. Let Φ : R n → C d × d be admissible ( see Definition . Then forall j ∈ { , . . . , n } , tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) = − tr b n d (cid:0) γ j,n Q ( R z Φ Q ) n − R z (cid:1) . (8.9) Proof.
One hastr b n d ( γ j,n Q R z CR z ) = tr b n d ( γ j,n Q R z ( Q Φ − Φ Q ) R z ) , = tr b n d ( γ j,n R z QQ Φ R z ) − tr b n d ( γ j,n Q R z Φ Q R z )= − tr b n d ( γ j,n Q R z Φ Q R z ) , using Proposition A.8 to deduce that tr b n d ( γ j,n R z QQ Φ R z ) = 0. In order toproceed to the proof of (8.9), we now show the following: For all odd k ∈ { , . . . , n } and ℓ ∈ { , . . . , k − } one hastr b n d (cid:0) γ j,n Q ( R z C ) k − R z (cid:1) = ( − ℓ tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ ( R z C ) k − − ℓ R z (cid:1) . (8.10)In the beginning of the proof we have dealt with the case k = 3. One notes thatequation (8.10) always holds for ℓ = 0. Next, we assume that k ∈ { , . . . , n } is odd,such that equality (8.10) holds for some ℓ ∈ { , . . . , k − } . Then one computestr b n d (cid:0) γ j,n Q ( R z C ) k − R z (cid:1) = ( − ℓ tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ ( R z C ) k − − ℓ R z (cid:1) = ( − ℓ tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ R z C ( R z C ) k − − ℓ − R z (cid:1) = ( − ℓ tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ R z ( Q Φ − Φ Q ) ( R z C ) k − − ( ℓ +1) R z (cid:1) = ( − ℓ tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ Q R z Φ ( R z C ) k − − ( ℓ +1) R z (cid:1) + ( − ℓ +1 tr b n d (cid:0) γ j,n Q ( R z Φ Q ) ℓ +1 ( R z C ) k − − ( ℓ +1) R z (cid:1) . By Corollary A.9, the first term on the right-hand side cancels, proving equation(8.10). Putting ℓ = k − (cid:3) The following result is needed for Lemma 8.10, however, it is also of independentinterest. Indeed, we will have occasion to use it rather frequently, when we discussthe case of three spatial dimensions specifically. Lemma 8.9 should be regardedas a regularization method, while preserving self-adjointness properties of the ( L -realization) of the underlying operators: Lemma 8.9.
Let ε > , n ∈ N , and T ∈ B (cid:0) H − ( n/ − ε ( R n ) , H ( n/ ε ( R n ) (cid:1) . Re-calling equation (5.3) , we consider t : R n × R n ∋ ( x, y ) (cid:10) δ { x } , T δ { y } (cid:11) . For µ > we denote T µ := (1 − µ ∆) − T (1 − µ ∆) − and t µ correspondingly. Then,for all ( x, y ) ∈ R n × R n , t µ ( x, y ) −→ µ ↓ t ( x, y ) . Proof.
It suffices to observe that for all s ∈ R , (1 − µ ∆) − −→ µ ↓ I strongly in H s ( R n )(see (5.1)). (cid:3) In order to proceed to prove the trace theorem, we need to investigate the as-ymptotic behavior of the integral kernel of J jL ( z ) given by (7.6) on the diagonal. ByProposition 7.4 together with Lemma 5.6, we can use Gauss’ divergence theoremfor computing the integral over the diagonal (see also (5.5)). Thus, in the expres-sion for the trace of χ Λ B L ( z ) we will use Gauss’ theorem for the ball centered at0 with radius Λ. Having applied the divergence theorem, we integrate over spheresof radius Λ. The volume element of the surface measure grows with Λ n − , so anyterm decaying faster than that will not contribute to the limit Λ → ∞ in (7.4).Consequently, any estimate of integral kernels (or differences of such) to follow withthe behavior of | x | n − γ for some γ > → ∞ , when computing the expression lim Λ →∞ tr( χ Λ B L ( z )). HE CALLIAS INDEX FORMULA REVISITED 65
Lemma 8.10.
Let n ∈ N odd, j ∈ { , . . . , n } , z ∈ C , Re( z ) > − and R z begiven by (4.6) as well as Q , C and γ j,n given by (6.3) , (6.15) and as in Remark . Then for n > , the integral kernel h ,j of b n d (cid:0) γ j,n Φ ( R z C ) n − R z (cid:1) satisfies, h ,j ( x, x ) = h ,j ( x, x ) + g ,j ( x, x ) , where h ,j is the integral kernel of b n d (cid:0) γ j,n Φ C n − R n z (cid:1) and g ,j satisfies forsome κ > , | g ,j ( x, x ) | κ (1 + | x | ) − n − ε , x ∈ R n , where ε > / is given as in Definition . In addition, if n > and z ∈ R , thenthe integral kernel h ,j of tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) vanishes on the diagonal.Proof. We discuss h ,j first and consider the operator B n := (Φ Q R z ) n − Φ = Φ ( Q R z Φ) n − , which is self-adjoint for all real z > −
1. Indeed, this follows from the self-adjointnessof Φ and the skew-self-adjointness of Q R z . For µ > B n,µ := (1 − µ ∆) − B n (1 − µ ∆) − . Then the integral kernel b n,µ of B n,µ is continuous. Moreover,for all real z > −
1, the opertor B n,µ is self-adjoint, by the self-adjointness of B n and so b n,µ is real and satisfies b n,µ ( x, y ) = b n,µ ( y, x ) for all x, y ∈ R n . By Lemma8.8 one recalls tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) = − tr b n d (cid:0) γ j,n Q ( R z Φ Q ) n − R z (cid:1) = − tr b n d (cid:0) γ j,n Q R z (Φ Q R z ) n − (cid:1) = − tr b n d (cid:0) γ j,n Q R z (Φ Q R z ) n − Φ Q R z (cid:1) = − tr b n d ( γ j,n Q R z B n Q R z ) . By Fubini’s theorem and the symmetry of B n,µ , one has for all j, k ∈ { , . . . , n } and x ∈ R n , z > − µ > x,µ ( j, k ) := ˆ R n × R n ( ∂ j r z )( x − x ) b n,µ ( x , x )( ∂ k r z )( x − x ) d n x d n x = ˆ R n × R n ( ∂ j r z )( x − x ) b n,µ ( x , x )( ∂ k r z )( x − x ) d n x d n x = ˆ R n × R n ( ∂ k r z )( x − x ) b n,µ ( x , x )( ∂ j r z )( x − x ) d n x d n x = Ψ x,µ ( k, j ) . By Lemma 8.9 one has for all x, y ∈ R n , h ,j ( x, y ) = − lim µ ↓ tr b n d (cid:18) n X i ,i =1 γ j,n γ i ,n γ i ,n ∂ i ˆ R n × R n r z ( x − x ) b n,µ ( x , x ) × ( ∂ i r z )( x − y ) d n x d n x (cid:19) = lim µ ↓ tr b n d (cid:18) n X i ,i =1 γ j,n γ i ,n γ i ,n ˆ R n × R n ( ∂ i r z )( x − x ) b n,µ ( x , x ) × ( ∂ i r z )( x − y ) d n x d n x (cid:19) . Thus, it follows from Corollary A.9 that h ,j ( x, x ) = lim µ ↓ tr b n d (cid:18) n X i ,i =1 γ j,n γ i ,n γ i ,n Ψ x,µ ( i , i ) (cid:19) = 0 , x ∈ R n . The assertion about h ,j is a direct consequence of Remark 5.18 and the asymptoticconditions imposed on Φ. (cid:3) For the estimate on the diagonal of the integral kernels of the operators underconsideration in the next theorem we need to choose the real part of z large. Infact, we use the Neumann series expression for the resolvents ( L ∗ L + z ) − and( LL ∗ + z ) − and Remark 5.15, both of which making the choice of large Re( z )necessary. We shall also have an a priori bound on the argument of z , recalling thedefinition (8.8) of the sector Σ z ,ϑ ⊂ C . Theorem 8.11.
Let L = Q + Φ be given by (7.1) , and for z ∈ C Re > − , let R z be given by (4.6) and C as in (6.15) . For j ∈ { , . . . , n } , let γ j,n ∈ C b n × b n ( cf.Remark , and ϑ ∈ (0 , π/ . Then there exists z > , such that for all z ∈ Σ z ,ϑ ( see (8.8)) , the integral kernels g ,j and g ,j of the operators tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) and tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) , respectively, satisfy for some κ > , (cid:2) | g ,j ( x, x ) | + | g ,j ( x, x ) | (cid:3) κ (1 + | x | ) − n , x ∈ R n . Proof.
We choose z such that √ z > n (one recalls Remark 5.15) and that for M := sup x ∈ R n k Φ( x ) k ∨ k ( Q Φ) ( x ) k one has 2 M [ z cos( ϑ )] − / /
2. We treat g ,j first. Let z ∈ Σ z ,ϑ , then, γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n = γ j,n Φ2 ( R z C ) n ∞ X k =0 ( R z C ) k R z = ∞ X k =0 γ j,n Φ2 ( R z C ) n ( R z C ) k R z . For x ∈ R n one infers (recalling δ { x } in (5.3)), g ,j ( x, x ) = (cid:28) δ { x } , ∞ X k =0 γ j,n Φ2 ( R z C ) k ( R z C ) n R z δ { x } (cid:29) = ∞ X k =0 (cid:10) δ { x } , γ j,n Φ2 ( R z C ) k ( R z C ) n R z δ { x } (cid:11) . HE CALLIAS INDEX FORMULA REVISITED 67
Hence, by Lemma 5.14 together with Remark 5.15, there exists c > x ∈ R n , (cid:12)(cid:12)(cid:10) δ { x } , γ j,n Φ2 ( R z C ) k ( R z C ) n R z δ { x } (cid:11)(cid:12)(cid:12) c (cid:18) M √ z (cid:19) k (cid:18)
11 + | x | (cid:19) n . Since 2 M [1 + z ] − / /
2, one concludes that | g ,j ( x, x ) | ∞ X k =0 (cid:12)(cid:12)(cid:12) h δ { x } , γ j,n Φ2 ( R z C ) k ( R z C ) n R z δ { x } i (cid:12)(cid:12)(cid:12) ∞ X k =0 c (cid:18) M √ z (cid:19) k (cid:18)
11 + | x | (cid:19) n c (cid:18)
11 + | x | (cid:19) n . The analogous reasoning applies to g ,j . (cid:3) We conclude the results on estimates of certain integral kernels on the diagonalwith the following corollary, which, roughly speaking, says that the diagonal of theintegral kernels involved is determined by the integral kernel of the operator to bediscussed in Proposition 8.13.
Corollary 8.12.
For z ∈ C , Re( z ) > − , denote R z as in (4.6) , let Φ : R n → C d × d be admissible ( see Definition , and L = Q + Φ as in (7.1) , ϑ ∈ (0 , π/ .In addition, denote J jL ( z ) for z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) as in (7.6) for all j ∈{ , . . . , n } , and C as in (6.15) . Moreover, let γ j,n ∈ C b n × b n , j ∈ { , . . . , n } ( cf.Remark . ( i ) Let n ∈ N > , j ∈ { , . . . , n } . Then there exists z > , such that if z ∈ Σ z ,ϑ ( see (8.8)) , and h and g denote the integral kernel of b n d (cid:0) γ j,n Φ C n − R n z (cid:1) and J jL ( z ) , respectively, then for some κ > , | h ( x, x ) − g ( x, x ) | κ (1 + | x | ) − n − ε , x ∈ R n , where ε > / is given as in Definition . ( ii ) The assertion of part ( i ) also holds for n = 3 , if, in the above statement, J jL ( z ) is replaced by J jL ( z ) − d ( γ j, Q R z CR z ) .Proof. One recalls from Lemma 7.7, J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) . With the help of Theorem 8.11 one deduces that the integral kernels of the last twoterms may be estimated by κ (1 + | x | ) − n on the diagonal. The integral kernel ofthe first term on the right-hand side vanishes on the diagonal, which is asserted inLemma 8.10. Hence, it remains to inspect the second term of the right-hand side.Thus, the assertion follows from Lemma 8.10. (cid:3) Having identified the integral kernel g j of 2 tr b n d (cid:0) γ j,n Φ C n − R n z (cid:1) to be the onlyterm determining the trace of χ Λ B L ( z ) as Λ → ∞ , we shall compute the integralover the diagonal of g j : Proposition 8.13.
Let n ∈ N > odd, C as in (6.15) , z ∈ C , Re( z ) > − , with R z given by (4.6) , Φ : R n → C d × d be admissible ( see Definition , γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , as in Remark . Then for j ∈ { , . . . , n } , the integral kernel g j of b n d (cid:0) γ j,n Φ C n − R n z (cid:1) satisfies, g j ( x, x ) = (1 + z ) − n/ (cid:18) i π (cid:19) ( n − / n − / × n X i ,...,i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1) , x ∈ R n , where ε ji ...i n − denotes the ε -symbol as in Proposition A.8 .Proof.
We recall that n = 2 b n + 1. With the help of Proposition A.8, g j is given by( x, y ) i ) b n n X i ...i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1) × ˆ ( R n ) n − r z ( x − x ) r z ( x − x ) · · · r z ( x n − − y ) d n x · · · d n x n − . Hence, by substitution in the integral expression and putting x = y , one obtains g j ( x, x ) = 2(2 i ) b n n X i ...i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1) × ˆ ( R n ) n − r z ( x ) r z ( x − x ) · · · r z ( x n − ) d n x · · · d n x n − . The last integral can be computed with the help of the Fourier transform and polarcoordinates, as was done in Proposition 5.8. In fact, one gets (see also [57, 3.252.2]), ˆ ( R n ) n − r z ( x ) r z ( x − x ) · · · r z ( x n − ) d n x · · · d n x n − = (2 π ) − n π n/ Γ ( n/ ˆ ∞ r n − r + 1 + z ) n dr = (2 π ) − n π n/ Γ ( n/
2) (1 + z ) − n/ − n √ π Γ ( n/ n − / n − π ( n − / n − / z ) − n/ , and notes that 2(2 i ) ( n − / n − π ( n − / n − / (cid:18) iπ (cid:19) ( n − / (4 n − − n +1) / n − / (cid:18) iπ (cid:19) ( n − / (3 n − / n − / (cid:18) i π (cid:19) ( n − / n − / . (cid:3) HE CALLIAS INDEX FORMULA REVISITED 69
Finally, we are ready to prove the (trace) Theorem 7.1, for n >
5, that is, weconsider the operator L = Q + Φ with an admissible potential Φ, such that Φ issmooth and attains values in the self-adjoint, unitary d × d -matrices. In addition, werecall that the first derivatives of Φ behave like | x | − for large x , whereas higher-order derivatives decay at least with the behavior | x | − − ε for large x and some ε > /
2. We note that we already established the Fredholm property of L inTheorem 6.3. We outline the proof of Theorem 7.1, for n >
5, as follows. The resultsin Section 7 yield the applicability of Theorem 3.4. More precisely, the operator χ Λ B L ( z ) is trace class with trace computable as the integral over the diagonal ofthe integral kernel of χ Λ B L ( z ). With Proposition 7.4 we will deduce that only theterm involving J jL ( z ), being analysed in Lemma 7.7, matters for the computation ofthe index. Next, we will show that { z tr( χ Λ B L ( z )) } Λ > is locally bounded usingLemma 5.6 (in particular (5.5)). The local boundedness result is then obtainedvia Gauss’ divergence theorem and Lemma 8.10 as well as Theorem 8.7. Havingproved local boundedness, we will use Montel’s theorem for deducing that at leastfor a sequence { Λ k } k ∈ N the limit f := lim k →∞ tr( χ Λ k B L ( · )) exists in the compactopen topology, that is, the topology of uniform convergence on compacts. With theresults from Corollary 8.12 and Proposition 8.13, choosing Re( z ) sufficiently large,we get an explicit expression for f . The explicit expression for f , by the principleof analytic continuation, carries over to z in a neighborhood of 0. As we know,by Theorem 3.4, that the limit lim Λ →∞ tr( χ Λ B L (0)) exists and coincides with theindex of L , we can then deduce that not only for the sequence { Λ k } k ∈ N but, in fact,the limit lim Λ →∞ tr( χ Λ B L ( · )) exists in the compact open topology and coincideswith f given in (7.4). The detailed arguments read as follows. Proof of Theorem for n > . By Theorem 7.8, χ Λ B L ( z ) is trace class for everyΛ >
0. Moreover, by Remark 7.9, tr( χ Λ B L ( z )) can be computed as the integral overthe diagonal of the respective integral kernel. Hence, by Proposition 7.4, equation(7.5), recalling also Remark 5.2, one obtains2 tr( χ Λ B L ( z )) = 2 ˆ B (0 , Λ) (cid:10) δ { x } , B L ( z ) δ { x } (cid:11) H − ( n/ − ε ,H ( n/ ε d n x = ˆ B (0 , Λ) (cid:28) δ { x } , (cid:18) n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) + A L ( z ) (cid:19) δ { x } (cid:29) d n x = ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) δ { x } (cid:29) d n x, (8.11)where we used Lemma 8.1 to deduce that (cid:10) δ { x } , A L ( z ) δ { x } (cid:11) = 0 for all x ∈ R n .Next, we prove that { z tr( χ Λ B L ( z )) } Λ > is locally bounded. One recalls fromLemma 7.7, J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) + tr b n d (cid:0) γ j,n Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:1) . Hence, n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) = n X j =1 (cid:2) ∂ j , (cid:0) b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + 2 tr b n d (cid:0) γ j,n Φ( R z C ) n − R z (cid:1)(cid:1)(cid:3) + tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) + tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) . (8.12)Denoting by h j the integral kernel of 2tr b n d (cid:0) γ j,n Φ C n − R n z (cid:1) , one observes thatfor some constant κ > | h j ( x, x ) | κ (1 + | x | ) − n , x ∈ R n . Hence, for any Λ > (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , b n d (cid:0) γ j,n Φ C n − R n z (cid:1)(cid:3) δ { x } (cid:29) d n x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (0 , Λ) n X j =1 ( ∂ j h j )( x, x ) d n x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ S n − n X j =1 h j ( x, x ) x j Λ d n − σ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ S n − n X j =1 | h j ( x, x ) | d n − σ ( x ) nκ (1 + Λ) − n Λ n − ω n − , (8.13)(with ω n − being the ( n − S n − = { x ∈ R n | | x | = 1 } , see (5.6)). The latter is uniformly bounded with respect toΛ > g ,j in that lemma as well as Gauss’ theorem,one arrives at (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1)(cid:3) δ { x } (cid:29) d n x + ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , (cid:0) b n d γ j,n Φ ( R z C ) n − R z (cid:1)(cid:3) δ { x } (cid:29) d n x − ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , b n d (cid:0) γ j,n Φ C n − R n z (cid:1) (cid:3) δ { x } (cid:29) d n x (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (0 , Λ) n X j =1 ( ∂ j g ,j )( x, x ) d n x (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ S n − n X j =1 g ,j ( x, x ) x j Λ d n − σ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ S n − n X j =1 | g ,j ( x, x ) | d n − σ ( x ) ˆ Λ S n − nκ (1 + | x | ) − n − ε d n − σ ( x ) nκ (1 + Λ) − n − ε ω n − Λ n − −→ Λ →∞ . (8.14) HE CALLIAS INDEX FORMULA REVISITED 71
Next, Theorem 8.7 implies that n z z tr (cid:16) χ Λ (cid:16) tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) + tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1)(cid:17)(cid:17)o Λ > (8.15)is bounded on any compact neighborhood of 0 intersected with B (0 , δ ) ∪ ( ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L )) for some δ >
0. Hence, summarizing equations (8.11) and (8.12), we getfor z ∈ C Re > − ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ): z χ Λ B L ( z )) = z ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) δ { x } (cid:29) d n x = z ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1)(cid:3) δ { x } (cid:29) d n x, + z ˆ B (0 , Λ) (cid:28) δ { x } , n X j =1 (cid:2) ∂ j , b n d (cid:0) γ j,n Φ( R z C ) n − R z (cid:1)(cid:3) δ { x } (cid:29) d n x + z ˆ B (0 , Λ) (cid:28) δ { x } , tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) δ { x } (cid:29) d n x + z ˆ B (0 , Λ) (cid:28) δ { x } , tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) δ { x } (cid:29) d n x = z ˆ B (0 , Λ) n X j =1 ( ∂ j g , j )( x, x ) d n x + z ˆ B (0 , Λ) n X j =1 ( ∂ j h j )( x, x ) d n x ++ z tr (cid:16) χ Λ (cid:16) tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1) + tr b n d (cid:0)(cid:2) Q , Φ (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR z ) n (cid:3)(cid:1)(cid:17)(cid:17) . Thus, with the estimates (8.13) and (8.14) together with (8.15), one infers that { z z tr( χ Λ B L ( z )) } Λ > is locally bounded on B (0 , δ ) ∪ C Re > for some δ >
0. By Lemma 8.3 together withTheorem 7.8, one infers that { z χ Λ B L ( z )) } Λ > is locally bounded on B (0 , δ ) ∪ C Re > . By Montel’s Theorem, there exists a sequence { Λ k } k ∈ N of positive reals tending to infinity such that { z χ Λ k B L ( z )) } k ∈ N converges in the compact open topology. We denote by f the respective limit. ThenLemma 5.6 implies that for k ∈ N ,2 tr( χ Λ k B L ( z )) = ˆ B (0 , Λ k ) n X j =1 ( ∂ j g j )( x, x ) d n x. and so f ( z ) = lim k →∞ ˆ B (0 , Λ k ) div G J,z ( x ) d n x. Here we denote G J,z := { x g j ( x, x ) } j ∈{ ,...,n } , with g j being the integral ker-nel of J jL ( z ) for j ∈ { , . . . , n } . Next, let ϑ ∈ (0 , π/
2) and choose z > i ). Let z ∈ Σ z ,ϑ , see (8.8). Recalling that h j is the integral ker-nel of 2tr b n d (cid:0) γ j,n Φ C n − R n z (cid:1) , we define H z := { x h j ( x, x ) } j ∈{ ,...,n } . Due toCorollary 8.12, one can find κ > k ∈ N , (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ k S n − (cid:18) ( G J,z − H z )( x ) , x Λ k (cid:19) R n d n − σ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ k S n − k ( G J,z − H z )( x ) k R n d n − σ ( x ) κ ˆ Λ k S n − (1 + | x | ) − n − ε d n − σ ( x )= κ Λ n − k ω n − (1 + Λ k ) − n − ε . Consequently, lim k →∞ ˆ Λ k S n − (cid:18) ( G J,z − H z )( x ) , x Λ k (cid:19) R n d n − σ ( x ) = 0 . Hence, with the help of Gauss’ theorem, f ( z ) = lim k →∞ ˆ B (0 , Λ k ) n X j =1 ( ∂ j g j )( x, x ) d n x = ˆ R n div G J,z ( x ) d n x = lim k →∞ ˆ B (0 , Λ k ) div G J,z ( x ) d n x = lim k →∞ ˆ Λ k S n − (cid:18) G J,z ( x ) , x Λ k (cid:19) R n d n − σ ( x )= lim k →∞ ˆ Λ k S n − (cid:18) H z ( x ) , x Λ k (cid:19) R n d n − σ ( x )= (cid:18) i π (cid:19) ( n − / n − / z ) − n/ lim k →∞ ˆ Λ k S n − × n X j =1 (cid:18) n X i ,...,i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1)(cid:19) × (cid:18) x j Λ k (cid:19) d n − σ ( x ) , (8.16)where, for the last integral, we used Proposition 8.13. By Theorem 3.4 one has f (0) = 2 ind( L ). In particular, any sequence { Λ k } k of positive reals converging toinfinity contains a subsequence { Λ k ℓ } ℓ such that for that particular subsequencethe limitlim ℓ →∞ ˆ Λ kℓ S n − n X j =1 (cid:18) n X i ,...,i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1)(cid:19) × (cid:18) x j Λ k ℓ (cid:19) d n − σ ( x ) HE CALLIAS INDEX FORMULA REVISITED 73 exists and equals 2 ind( L ) [( n − / i/ (8 π )] ( n − / . (8.17)Hence, the limitlim Λ →∞ ˆ Λ S n − n X j =1 (cid:18) n X i ,...,i n − =1 ε ji ...i n − tr (cid:0) Φ( x ) ( ∂ i Φ) ( x ) . . . (cid:0) ∂ i n − Φ (cid:1) ( x ) (cid:1) (cid:19) × (cid:16) x j Λ (cid:17) d n − σ ( x ) (8.18)exists and equals the number in (8.17). On the other hand, for z ∈ Σ z ,ϑ , (see againCorollary 8.12) the family { z tr( χ Λ B L ( z )) } Λ > converges for Λ → ∞ on the domain Σ z ,ϑ if and only if the limit in (8.18) exists.Indeed, this follows from the explicit expression for the limit in (8.16). Therefore, { z tr( χ Λ B L ( z )) } Λ > converges in the compact open topology on Σ z ,ϑ . By the local boundedness of thelatter family on the domain B (0 , δ ) ∪ C Re > , the principle of analytic continuationimplies that the latter family actually converges on the domain B (0 , δ ) ∪ C Re > inthe compact open topology. In particular,2 f ( z )(1 + z ) n/ [( n − / i/ (8 π )] ( n − / = lim Λ →∞ ˆ Λ S n − n X j =1 (cid:18) n X i ,...,i n − =1 ε ji ...i n − tr (cid:0) Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x ) (cid:1)(cid:19) × (cid:18) x j Λ (cid:19) d n − σ ( x ) . (cid:3) The Case n = 3In this section we shall discuss the necessary modifications, such that Theorem7.1 continues to hold also for the case n = 3. The main issue for the need of extraarguments for this case is the lack of differentiability of the integral kernel of J jL ( z )given in Lemma 8.1. The main issue being the first summand in the expression for J jL ( z ) derived in Lemma 7.7, that is, the termtr b n d (cid:0) γ j,n Q ( R z C ) R z (cid:1) , for the integral kernel of which we fail to show differentiability. Indeed, as thisopertor increases regularity only by 3 orders of differentiability, not even continuityof the associated integral kernel is clear. The basic idea to overcome this difficultyand to get the result asserted in Theorem 7.1 also for the case n = 3 has alreadybeen used and is contained in Lemma 8.9. So, the operator B L ( z ) will be multipliedfrom the left and from the right by (1 − µ ∆) − for some µ >
0. The reason formultiplying from both sides is that we wanted to re-use strategies for showing thatcertain integral kernels vanish on the diagonal. The key for the latter argumentshas been the self-adjointness of the operators under consideration, which, in turn,result in symmetry properties for the associated integral kernel.An additional fact, enabling the strategy just sketched for the case n = 3, is thefollowing result. Proposition 9.1 (See, e.g., [92], p. 28–29, or [105], Lemma 6.1.3) . Assume H is acomplex, separable Hilbert space, B ∈ B ( H ) , and A > is self-adjoint in H . Thenfor µ > , B µ := (1 + µA ) − B (1 + µA ) − ∈ B ( H ) and B µ → B in B ( H ) as µ ↓ .In particular, tr H ( B µ ) −→ µ ↓ tr H ( B ) . Next, we will give the details for the modifications of the proof of Theorem 7.1for the case n = 3. Thus, for µ >
0, we introduce the operator B L,µ ( z ) := (1 − µ ∆) − B L ( z ) (1 − µ ∆) − , (9.1)where z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ), L = Q + Φ given by (7.1), and B L ( z ) is given by(7.2). We also introduce J jL,µ ( z ) = (1 − µ ∆) − (cid:0) tr b n d (cid:0) L (cid:0) L ∗ L + z ) − γ j,n (cid:1) + tr b n d (cid:0) L ∗ ( LL ∗ + z ) − γ j,n (cid:1)(cid:1) (1 − µ ∆) − , (9.2)with γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , as in Remark 6.1, and A L,µ ( z ) = (1 − µ ∆) − (cid:0) tr b n d (cid:0)(cid:2) Φ , L ∗ ( LL ∗ + z ) − (cid:3)(cid:1) − tr b n d (cid:0)(cid:2) Φ , L ( L ∗ L + z ) − (cid:3)(cid:1)(cid:1) (1 − µ ∆) − (9.3)for the admissible potential Φ (see Definition 6.11). By Theorem 7.8 and theideal property of B ( H ), the operator B L,µ ( z ) is trace class for all µ > z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) and Re( z ) > −
1. As for the case n >
5, we need thefollowing more detailed description of the operator B L,µ ( z ): Lemma 9.2.
Let L = Q + Φ as in (7.1) , µ > , z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) , with Re( z ) > − . Then with J jL,µ ( z ) , j ∈ { , . . . , m } , and A L,µ ( z ) given by (9.2) and (9.3) , one has B L,µ ( z ) = n X j =1 (cid:2) ∂ j , J jL,µ ( z ) (cid:3) + A L,µ ( z ) . HE CALLIAS INDEX FORMULA REVISITED 75
Proof.
The only nontrivial item to be established, invoking Proposition 7.4 togetherwith equations (7.5), (7.6), and (7.7), is to establish that for j ∈ { , . . . , n } , (cid:2) ∂ j , J jL,µ ( z ) (cid:3) = (1 − µ ∆) − (cid:2) ∂ j , J jL ( z ) (cid:3) (1 − µ ∆) − . Recalling γ j,n ∈ C b n × b n , j ∈ { , . . . , n } (cf. Remark 6.1), one observes that(1 − µ ∆) − (cid:2) ∂ j , tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1)(cid:3) (1 − µ ∆) − = (1 − µ ∆) − ∂ j tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) − tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) ∂ j (1 − µ ∆) − = ∂ j (1 − µ ∆) − tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) (1 − µ ∆) − − (1 − µ ∆) − tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) (1 − µ ∆) − ∂ j = (cid:2) ∂ j , (1 − µ ∆) − tr b n d (cid:0) L ( L ∗ L + z ) − γ j,n (cid:1) (1 − µ ∆) − (cid:3) , yielding the assertion. (cid:3) In contrast to the operator J jL ( z ), the integral kernel for the regularized operator J jL,µ ( z ) satisfies the desired differentiability properties: Corollary 9.3.
Let L = Q + Φ be given by (7.1) , z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) , with Re( z ) > − , and suppose µ > . If n ∈ N is odd, then for all j ∈ { , . . . , n } , theintegral kernel of J jL,µ ( z ) given by (9.2) is continuously differentiable.Proof. We recall R z as given by (4.6), Q and C given by (6.3) and (6.15), respec-tively, as well as γ j,n as in Remark 6.1. According to Proposition 5.4 for ℓ ∈ R , itsuffices to observe that the operator(1 − µ ∆) − tr b n d γ j,n Q ( R z C ) n − R z (1 − µ ∆) − is continuous from H ℓ ( R n ) (see (5.1)) to H ℓ +2( n − − ( R n ) = H ℓ +2 n +1 ( R n ) . Thus, by Corollary 5.3, the assertion follows from 2 n > n . (cid:3) Next, we turn to a variant of the first assertion in Lemma 8.10.
Lemma 9.4.
Let µ > , z ∈ C , Re( z ) > − , R z given by (4.6) , C given by (6.15) , and Q given by (6.3) . Then for all j ∈ { , , } , the integral kernel of (1 − µ ∆) − tr d ( γ j, Q R z CR z ) (1 − µ ∆) − vanishes on the diagonal, where γ , , γ , , γ , ∈ C × are given as in Remark see also Appendix A) .Proof. We denote the integral kernel under consideration by h j , j ∈ { , , } . FromLemma 8.8, one recalls,(1 − µ ∆) − tr d ( γ j, Q R z CR z ) (1 − µ ∆) − = − (1 − µ ∆) − tr d ( γ j, Q R z Φ Q R z ) (1 − µ ∆) − . With (1 − µ ∆) − = (1 /µ ) ((1 /µ ) − ∆) − one computes, h j ( x, x ) = − µ ˆ ( R ) r /µ ( x − x ) tr d (cid:18) γ j, X i =1 γ i , ( ∂ i r z )( x − x )Φ( x ) × X i =1 γ i , ( ∂ i r z )( x − x ) (cid:19) r /µ ( x − x ) d n x d n x d n x = 1 µ ˆ ( R ) r /µ ( x − x ) tr d (cid:18) γ j, X i =1 γ i , ( ∂ i r z )( x − x )Φ( x ) × X i =1 γ i , ( ∂ i r z )( x − x ) (cid:19) r /µ ( x − x ) d n x d n x d n x = 1 µ ˆ ( R ) r /µ ( x − x ) tr d (cid:18) γ j, X i =1 γ i , ( ∂ i r z )( x − x )Φ( x ) × X i =1 γ i , ( ∂ i r z )( x − x ) (cid:19) r /µ ( x − x ) d n x d n x d n x , x ∈ R n . The latter expression is symmetric in x and x , by Fubini’s theorem. Hence, theassertion follows as in Lemma 8.8 with the help of Corollary A.9. (cid:3) Now we are in position to prove the trace theorem for dimension n = 3. Of coursethe principal strategy for the proof is similar to the one for dimensions n > Theorem 9.5.
Let n = 3 , L = Q + Φ given by (7.1) , and χ Λ given by (7.3) .Then for all z ∈ C with z ∈ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , and B L ( z ) given (7.2) , χ Λ B L ( z ) is trace class for all Λ > . The limit f ( · ) := lim Λ →∞ tr( χ Λ B L ( · )) exists in thecompact open topology and the formula f ( z ) = i π (1 + z ) − / lim Λ →∞ X j,i ,i =1 ε ji i × ˆ Λ S tr (cid:0) Φ( x ) ( ∂ i Φ) ( x ) ( ∂ i Φ) ( x ) (cid:1) x j d n − σ ( x ) (9.4) holds, where ε ji i denotes the ε -symbol as in Proposition A.8 .Proof.
Let Λ > µ >
0. Denote the integral kernels of A L ( z ) and P j (cid:2) ∂ j , J jL ( z ) (cid:3) by A L and J L , respectively, and correspondingly for A L,µ ( z ) and P j (cid:2) ∂ j , J jL,µ ( z ) (cid:3) ,where the respective operators are given by (7.7), (7.6), (9.3), and (9.2). One notesthat by Lemma 8.9, A L,µ → A L and J L,µ → J L pointwise as µ →
0. One recallsfrom Proposition 7.4 and Theorem 7.8 together with Proposition 4.3 that (similarlyto the case n = 5),2 tr( χ Λ B L ( z )) = ˆ B (0 , Λ) ( A L + J L ) ( x, x ) d n x, and, as A L and the integral kernel of B L ( z ) are continuous, so is J L . Hence, byLemma 8.9 and using A L ( x, x ) = 0 (see Lemma 8.1), one obtains2 tr( χ Λ B L ( z )) = ˆ B (0 , Λ) A L ( x, x ) + J L ( x, x ) d n x = ˆ B (0 , Λ) lim µ → J L,µ ( x, x ) d n x HE CALLIAS INDEX FORMULA REVISITED 77 = lim µ → ˆ B (0 , Λ) J L,µ ( x, x ) d n x, where the last equality follows from the fact that the family of integral kernels of { B L,µ ( z ) − A L,µ ( z ) } µ> is locally uniformly bounded: To prove the latter assertion, we note that due toCorollary 5.3, { B L,µ ( z ) − A L,µ ( z ) } µ> defines a uniformly bounded family of con-tinuous linear operators from H ℓ ( R n ) (see (5.1)) to H ℓ +2 n − ( R n ), ℓ ∈ R . Indeed,this follows from the representation in Lemma 7.7 together with Proposition 5.4and the fact that for all s ∈ R , (1 − µ ∆) − → I strongly in H s ( R n ). Next, wedenote K L,µ := (cid:8) x g jL,µ ( z )( x, x ) (cid:9) j ∈{ , , } , where g jL,µ ( z ) is the integral kernel of J jL,µ ( z ), j ∈ { , , } , and K L that for (cid:8) J jL ( z ) − d ( γ j, Q R z CR z ) (cid:9) j ∈{ , , } . Invoking Lemmas 9.4 and 8.9, and hence the fact that { x K L,µ ( x ) } µ> is locallyuniformly bounded, one obtainslim µ → ˆ B (0 , Λ) J L,µ ( x, x ) d n x = lim µ → ˆ Λ S (cid:16) K L,µ ( x ) , x Λ (cid:17) d n − σ ( x )= ˆ Λ S lim µ → (cid:16) K L,µ ( x ) , x Λ (cid:17) d n − σ ( x )= ˆ Λ S (cid:16) K L ( x ) , x Λ (cid:17) d n − σ ( x ) . As in the case n >
5, one computes with the help of Corollary 8.12 that for z ∈ Σ z ,ϑ , see (8.8), for some fixed ϑ ∈ (0 , π/
2) and z ∈ R sufficiently large, thelimit Λ → ∞ actually coincides with K L replaced by the vector of integral kernelsof (cid:8) d (cid:0) γ j, Φ C R z (cid:1)(cid:9) j ∈{ , , } (employing analogous arguments using Lemmas8.5, 8.6, and 8.3). Hence one can compute this expression explicitly with the helpof Proposition 8.13, ending up with (9.4). (cid:3) The Index Theorem and Some Consequences
Putting the results of the Sections 3 and 7 together, we arrive at the followingtheorem:
Theorem 10.1.
Let n ∈ N > odd, d ∈ N , Φ : R n → C d × d admissible ( see Definition . Then the operator L = Q + Φ given by (7.1) is Fredholm and ind( L ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr(Φ( x )( ∂ i Φ)( x ) . . . ( ∂ i n − Φ)( x )) x j d n − σ ( x ) , (10.1) where ε ji ...i n − denotes the ε -symbol as introduced in Proposition A.8 .Proof.
Appealing to Theorem 3.4 and Theorem 7.1 (or 9.5), we have f (0) = ind( L ),with f from Theorem 7.1. (cid:3) In Corollary 10.11 at the end of this section we will show that actually, ind( L ) = 0for admissible Φ.Next, we indicate how the index theorem obtained can be generalized to poten-tials Φ belonging only to C b (cid:0) R n ; C d × d (cid:1) satisfying | Φ( x ) | > c for all x ∈ R n \ B (0 , R )for some R > c >
0. More precisely, we will prove the following theorem later inSection 12 in the case where Φ is C ∞ and in full generality in Section 13: Theorem 10.2.
Let n ∈ N > odd, d ∈ N , Φ ∈ C b (cid:0) R n ; C d × d (cid:1) . Assume the followingproperties Φ( x ) = Φ( x ) ∗ , x ∈ R n , there exists c > , R > such that | Φ( x ) | > c , x ∈ R n \ B (0 , R ) , and that there is ε > / such that for all α ∈ N n , | α | < , there is κ > such that k ( ∂ α Φ)( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | = 2 , x ∈ R n . We recall Q = P nj =1 γ j,n ∂ j , with γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , given in (6.3) orTheorem . Then the operator L := Q + Φ considered in L ( R n ) b n d is a Fredholmoperator and ind( L ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − (10.2) × ˆ Λ S n − tr( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x j d n − σ ( x ) , where U ( x ) = | Φ( x ) | − Φ( x ) = sgn(Φ( x )) , x ∈ R . While in this manuscript we focus on the functional analytic proof of Callias’index formula (10.2), we refer to the discussion by Bott and Seeley [14] for itsunderlying topological setting (homotopy invariants, etc.).In Theorem 10.2, there are two main difficulties to cope with: on the one hand– in contrast to the situation in Theorem 10.1 – the potential is only assumed tobe invertible on the complement of large balls, on the other hand the potential
HE CALLIAS INDEX FORMULA REVISITED 79 is only C . We will address the second case later on, and concern ourselves withthe invertibility issue first. However, before providing the proof of Theorem 10.2in these more general cases, we give a motivating fact underlining the need forTheorem 10.2. In particular, in Theorem 10.6, we show that a particular class ofpotentials cannot be treated with the help of Theorem 10.1. The main problempreventing the applicability of Theorem 10.1 is the everywhere invertibility assumedin Definition 6.11.We note that the special case n = 3 in connection with Yang–Mills–Higgs fieldsand monopoles has been discussed in detail in [70, Sect. II.5] and [81, Sect. VIII.4].Before turning to Theorem 10.6, we shall provide a result that studies the signof an operator. This study is needed, as the formula in Theorem 10.2 involves x Φ( x ) / | Φ( x ) | = sgn(Φ( x )). Theorem 10.3.
Let H be a Hilbert space, A ∈ B ( H ) , Re (cid:0) A (cid:1) > c for some c > .Then the integral ( see, e.g., [59, Ch. 5, equation (5.3)]) , sgn( A ) := 2 π A ˆ ∞ (cid:0) t + A (cid:1) − dt converges in B ( H ) and sgn( · ) is analytic on B (cid:0) A, ( k A k + c ) / − k A k ) (cid:1) . Moreover,if in addition, A = A ∗ then sgn( A ) = A | A | − , and sgn( A ) is unitary.Proof. From Re (cid:0) t + A (cid:1) > t + c it follows that (cid:13)(cid:13) (cid:0) t + A (cid:1) − (cid:13)(cid:13) ( t + c ) − and,thus, the integral converges in operator norm. In order to prove analyticity, we firstshow that given B ∈ B ( H ) with Re( B ) > c the function T ´ ∞ (cid:0) t + B + T (cid:1) − dt is analytic at 0 with convergence radius at least c . Hence, let B ∈ B ( H ) withRe( B ) > c. Then (cid:13)(cid:13) (cid:0) t + B (cid:1) − (cid:13)(cid:13) ( t + c ) − c − for all t ∈ R . If T ∈ B ( H ) with k T k ϑc for some 0 < ϑ <
1, then (cid:13)(cid:13)(cid:13)(cid:0) t + B (cid:1) − T (cid:13)(cid:13)(cid:13) ϑ for all t ∈ R and thus, ˆ ∞ (cid:0) t + ( B + T ) (cid:1) − dt = ˆ ∞ (cid:16) (cid:0) t + B (cid:1) − T (cid:17) − (cid:0) t + B (cid:1) − dt = ˆ ∞ ∞ X k =0 (cid:16) − (cid:0) t + B (cid:1) − T (cid:17) k (cid:0) t + B (cid:1) − dt = ∞ X k =0 ˆ ∞ (cid:16) − (cid:0) t + B (cid:1) − T (cid:17) k (cid:0) t + B (cid:1) − dt. One observes that c k : B ( H ) k → B ( H ) given by c k ( T, . . . , T ) := ˆ ∞ (cid:16) − (cid:0) t + B (cid:1) − T (cid:17) k (cid:0) t + B (cid:1) − dt, is a bounded k -linear form with bound c − k π/ (2 √ c ). Indeed, for the contractions T , . . . , T k ∈ B ( H ) one estimates (cid:13)(cid:13)(cid:13)(cid:13) ˆ ∞ k Y j =1 (cid:2) − ( t + B ) − T j (cid:3) ( t + B ) − dt (cid:13)(cid:13)(cid:13)(cid:13) ˆ ∞ k Y j =1 (cid:13)(cid:13)(cid:2) − ( t + B ) − T j (cid:3) ( t + B ) − (cid:13)(cid:13) dt ˆ ∞ (cid:13)(cid:13) ( t + B ) − k (cid:13)(cid:13)(cid:13)(cid:13) ( t + B ) − (cid:13)(cid:13) dt (cid:18) c (cid:19) k ˆ ∞ t + c dt = (cid:18) c (cid:19) k π √ c . In particular, the power series has convergence radius at least c . It follows that T ´ ∞ (cid:0) t + T (cid:1) − dt is analytic about A with convergence radius c. If T ∈ B ( H )with k T k < (cid:0) ( k A k + c ) / − k A k (cid:1) , then (cid:13)(cid:13) ( A + T ) − A (cid:13)(cid:13) < k A kk T k + k T k k A k (cid:0) ( k A k + c ) / − k A k (cid:1) + (cid:0) ( k A k + c ) / − k A k (cid:1) = c. Hence, the map T ´ ∞ (cid:0) t + T (cid:1) − dt is analytic about A with convergence radiusat least (cid:0) ( k A k + c ) / − k A k (cid:1) .The equality and unitarity now follow from the functional calculus for self-adjointopertors and the respective equality for numbers. (cid:3) The next fact provides a more detailed account on the behavior of x sgn(Φ( x ))for smooth Φ. We note that the following result has been asserted implicitly in amodified form in [22, last paragraph on p. 226]. Lemma 10.4.
Let n, d ∈ N > , Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) pointwise self-adjoint, c, R > , c = 0 . Assume that for all x ∈ R n \ B (0 , R ) , | Φ( x ) | > c . Let τ > . Then thereexists U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) pointwise self-adjoint, and a function u ∈ C ∞ b ( R n ; R > ) with u , u R n \ B (0 ,τ ) = 1 , such that U ( x ) = sgn(Φ( x )) , x ∈ R n \ B (0 , R ) and U ( x ) = u ( x ) I d , x ∈ R n . Moreover, for all β ∈ N n , β = 0 , there exists κ > such that for all x ∈ R n \ B (0 , R ) , k ∂ β U ( x ) k κ X α ∈ N n ,γ ∈ N , | α | γ = | β | k ∂ α Φ( x ) k γ . Remark . ( i ) We note that the function U constructed in Lemma 10.4 attainsvalues in the set of unitary matrices (on R n \ B (0 , R )). Indeed, this follows fromTheorem 10.3.( ii ) In the situation of Lemma 10.4, assume, in addition, that Φ satisfies the fol-lowing estimates: For some ε > / α ∈ N n , there is a constant κ > k ( ∂ α Φ)( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | > , x ∈ R n . Then U constructed in Lemma 10.4 satisfies analogous estimates: For α ∈ N n thereexists κ > k ( ∂ α U )( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | > . In particular, if Φ is admissible (see Definition 6.11), then so is U = sgn(Φ). ⋄ HE CALLIAS INDEX FORMULA REVISITED 81
Proof of Lemma . One observes that x sgn(Φ( x )) is C ∞ b on R n \ B (0 , R ′ ) forsome 0 < R ′ < R , by Theorem 10.3. Moreover, for j ∈ { , . . . , n } ,( ∂ j U )( x ) = (sgn ′ (Φ( x )))( ∂ j Φ)( x ) . Thus, ( ∂ k ∂ j U )( x ) = sgn ′′ (Φ( x ))( ∂ k Φ)( x )( ∂ j Φ)( x ) + sgn ′ (Φ( x ))( ∂ k ∂ j Φ)( x ) . Continuing in this manner, we obtain the estimates for the derivatives, once noticingthat x sgn ( k ) (Φ( x )) is bounded for all k ∈ N . Indeed, by the boundedness ofΦ and since | Φ( x ) | > c for all x ∈ R n \ B (0 , R ), the set { Φ( x ) | x ∈ R n \ B (0 , R ) } ⊆ C d × d is relatively compact and its closure is contained in the domain of analyticityof sgn( · ). Hence, x sgn ( k ) (Φ( x )) is indeed bounded.Next, let η ∈ C ∞ b ( R n ) with η ( x ) = R ′ , | x | R ′ , ∈ [ R ′ , R ] , R ′ < | x | < R, = | x | , | x | > R. Then α : R n \{ } → R n , x η ( | x | ) x | x | is C ∞ and α ( x ) = x for all | x | > R . Let, inaddition, φ : R n → R > be a C ∞ -function such that φ ( x ) = 1 for x ∈ R n \ B (0 , τ )and with 0 φ φ ( x ) = 0 on B (0 , τ / U is x φ ( x ) sgn(Φ( α ( x ))) . (cid:3) One might wonder, whether the function u vanishing at the origin in Lemma 10.4is really needed. In fact, if it was possible for any arbitrarily differentiable potentialΦ discussed in Theorem 10.2, to choose u in Lemma 10.4 being 1 also at the origin,the only nontrivial assertion of Theorem 10.2 would be the differentiability issue.However, the next example indicates that Theorem 10.2 has a nontrivial application. Theorem 10.6.
Consider the function
Φ : R → C × such that Φ( x ) = X j =1 σ j x j | x | , | x | > , as in Example . Then there is no U ∈ C ∞ (cid:0) R ; C × (cid:1) with the property that U ( x ) = Φ( x ) for all x ∈ R n , | x | > , U ( x ) = U ( x ) ∗ and for some c > , | U ( x ) | > c , x ∈ R n .Proof. We will proceed by contradiction and assume the existence of such a U . ByLemma 10.4 (and Remark 10.5 ( i )), we may assume without loss of generality that U assumes values in the self-adjoint unitary operators in C × . The latter are ofthe form W = (cid:18) a b + idb − id c (cid:19) , a, b, c, d ∈ R with W ∗ W = I . From the latter equation, one reads off1 = a + b + d , a + c ) b, a + c ) d, c + b + d . Hence, either a = − c , which implies b = d = 0 and a = c = ±
1, or a = − c with a + b + d = 1. Note that, in the latter case, we have W = aσ + bσ + dσ anddet( W ) = −
1. Hence, since U is pointwise invertible everywhere, and det( ± I ) = 1,by the intermediate value theorem, one infers that U [ R ] ⊆ { aσ + bσ + cσ | a, b, c ∈ R , a + b + c = 1 } =: U Identifying U with S and using U | S = I S , one observes that U is a retractionof B (0 ,
1) for S , which is a contradiction. We provide some details for the latterclaim. Assume there exists a continuous map f : B (0 , ⊂ R → S with theproperty f ( x ) = x for all x ∈ S . Denoting the identity on B (0 ,
1) by I B (0 , , oneconsiders the homotopy H of f and I B (0 , given by H ( λ, x ) := λf ( x ) + (1 − λ ) I B (0 , ( x ) , λ ∈ [0 , , x ∈ B (0 , . In the following, we denote by deg( g, z ) Brouwer’s degree of a function g : B (0 , → R in the point z ∈ R \ g [ S ]. One observes that 0 ∈ R \ H ( λ, S ) = R \ S forall λ ∈ [0 , f . Hence, by homotopy invariance of Brouwer’sdegree, one gets, using 0 ∈ I B (0 , [ B (0 , / ∈ f [ B (0 , S ,1 = deg( I B (0 , ,
0) = deg( H (0 , · ) ,
0) = deg( H (1 , · )) = deg( f,
0) = 0 , a contradiction. (cid:3) While we decided to provide an explicit proof of Theorem 10.6, it should bementioned that is is a special case of “Brouwer’s no retraction theorem” (see, e.g.,[38, Theorem 3.12]): There is no continuous map f : B (0 , → S n − that is theidentity on S n − . (Here B (0 ,
1) denotes the closed unit ball in R n , n ∈ N .)In the remainder of this section, we study the index formula (10.2) in more detail.More precisely, we will show an invariance principle which will lead to a proof ofCorollary 10.11, which shows that for admissible potentials Φ, the index of Q + Φvanishes, reproducing [86, Theorem 5.2] in our context.Let n, d ∈ N , U ⊆ R n open, Υ ∈ C (cid:0) U ; C d × d (cid:1) . For x ∈ U we introduce theexpression M Υ ( x ) := n X i ,...,i n =1 ε i ··· i n tr (cid:0) ∂ i Υ( x ) · · · ∂ i n Υ( x ) (cid:1) , (10.3)where ε i ··· i n denotes the totally anti-symmetric symbol in n coordinates. Remark . The relationship of the index formula for potentials Φ as in Theorem10.2 and the function defined in (10.3) is as follows: Let U be C -smooth with U = sgn(Φ) on the complement of a sufficiently large ball. For Λ >
0, one computeswith the help of Gauss’ divergence theorem1Λ n X i ,...,i n =1 ε i ...i n ˆ Λ S n − tr( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x i n d n − σ ( x )= n X i ,...,i n =1 ε i ...i n ˆ B (0 , Λ) tr(( ∂ i U )( x ) . . . ( ∂ i n U )( x )) d n x = ˆ B (0 , Λ) M U ( x ) d n x. HE CALLIAS INDEX FORMULA REVISITED 83
Hence, the index formula for the operator L = Q + Φ discussed in Theorem 10.2may be rewritten as followsind( L ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ ˆ B (0 , Λ) M U ( x ) d n x. (10.4) ⋄ Definition 10.8 (Transformations of constant orientation) . Let n ∈ R n , U ⊆ R n open, dense. We say that T : U → R n is a transformation of constant orientation ,if the following properties ( i ) – ( iii ) are satisfied: ( i ) T is continuously differentiable and injective. ( ii ) T [ U ] is dense in R n . ( iii ) The function
U ∋ x sgn(det( T ′ ( x ))) is either identically or − . We define sgn( T ) := sgn(det( T ′ ( x ))) for some ( and hence for all ) x ∈ U . The sought after invariance principle then reads as follows:
Theorem 10.9.
Let n ∈ N > odd, d ∈ N , Φ ∈ C b (cid:0) R n ; C d × d (cid:1) . Assume the followingproperties: Φ( x ) = Φ( x ) ∗ , x ∈ R n , there exists c > , R > such that | Φ( x ) | > c for all x ∈ R n \ B (0 , R ) , and thatthere is ε > / such that for all α ∈ N n , | α | < , there is κ > such that k ( ∂ α Φ)( x ) k ( κ (1 + | x | ) − , | α | = 1 ,κ (1 + | x | ) − − ε , | α | = 2 , x ∈ R n . We recall Q = P nj =1 γ j,n ∂ j , with γ j,n ∈ C b n × b n , j ∈ { , . . . , n } , given in (6.3) orin Theorem . In addition, let T : U ⊆ R n → R n ( with U as in Definition be a transformation of constant orientation. Assume that Φ T := Φ ◦ T ( the closureof the mapping Φ ◦ T ) satisfies the assumptions imposed on Φ . Then L = Q + Φ and L = Q + Φ T are Fredholm and ind( L ) = sgn( T ) ind( L ) . Before proving Theorem 10.9 we need a chain rule for the function defined in(10.3).
Lemma 10.10.
Let n, d ∈ N , U ⊆ R n open, Φ ∈ C (cid:0) R n ; C d × d (cid:1) , T ∈ C ( U ; R n ) .Then, M Φ ◦ T ( x ) = M Φ ( T ( x )) det( T ′ ( x )) , x ∈ U . Proof.
One recalls that for an n × n -matrix A = ( a ij ) i,j ∈{ ,...,n } ∈ C n × n , its deter-minant may be computed as followsdet( A ) = n X i ,...,i n =1 ε i ··· i n a i · · · a i n n . Consequently, for k , . . . , k n ∈ { , . . . , n } , one gets ε k ··· k n det( A ) = n X i ,...,i n =1 ε i ··· i n a i k · · · a i n k n . Using the chain rule of differentiation, one obtains for x ∈ U , M Φ ◦ T ( x ) = n X i ,...,i n =1 ε i ··· i n tr d (cid:0) ∂ i (Φ ◦ T )( x ) · · · ∂ i n (Φ ◦ T )( x ) (cid:1) = n X i ,...,i n =1 ε i ··· i n tr d (cid:0) n X k =1 ∂ k Φ( T ( x )) ∂ i T k ( x ) · · · n X k n =1 ∂ k n Φ( T ( x )) ∂ i n T k n ( x ) (cid:1) = n X k ,...,k n =1 n X i ,...,i n =1 ε i ··· i n ∂ i T k ( x ) · · · ∂ i n T k n ( x ) × tr d (cid:0) ∂ k Φ( T ( x )) · · · ∂ k n Φ( T ( x )) (cid:1) = n X k ,...,k n =1 ε k ··· k n det( T ′ ( x )) tr d (cid:0) ∂ k Φ( T ( x )) · · · ∂ k n Φ( T ( x )) (cid:1) = M Φ ( T ( x )) det( T ′ ( x )) . (cid:3) Proof of Theorem . Let U be C -smooth and such that sgn(Φ) = U on com-plements of sufficiently large balls. One observes that U T := U ◦ T = sgn(Φ T ). Inparticular, ind( Q + Φ T ) = ind( Q + U T ). Next, we set c n := 12 (cid:18) i π (cid:19) ( n − / n − / . By Theorem 10.2 together with Remark 10.7, and taking into account the chainrule, Lemma 10.10, one computes,ind( Q + U T ) = c n lim Λ →∞ ˆ B (0 , Λ) M U T ( x ) d n x = c n lim Λ →∞ ˆ B (0 , Λ) M U ◦ T ( x ) d n x = c n lim Λ →∞ ˆ B (0 , Λ) M U ( T ( x )) det( T ′ ( x )) d n x = sgn( T ) c n lim Λ →∞ ˆ B (0 , Λ) M U ( T ( x )) | det( T ′ ( x )) | d n x = sgn( T ) c n lim Λ →∞ ˆ T [ B (0 , Λ)] M U ( x ) d n x = sgn( T ) c n lim Λ →∞ ˆ B (0 , Λ) ∩ T [ B (0 , Λ)] M U ( x ) d n x, using the transformation rule for integrals.To conclude the proof, we are left with showing c n lim Λ →∞ ˆ B (0 , Λ) ∩ T [ B (0 , Λ)] M U ( x ) d n x = ind( Q + U ) . For this purpose one notes that T is continuously invertible, by hypothesis. Hence,the range of T is open. Since the range of T is also dense, { χ T [ B (0 , Λ)] } Λ ∈ N convergesin the strong operator topology of B (cid:0) L ( R n ) (cid:1) to I L ( R n ) , where χ T [ B (0 , Λ)] denotesthe characteristic function of the set T [ B (0 , Λ)], Λ >
0. Thus, for L = Q + U , one HE CALLIAS INDEX FORMULA REVISITED 85 computesind( L ) = lim Λ →∞ lim z → + z tr L ( R n ) (cid:0) χ T [ B (0 , Λ)] χ Λ tr b n d (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1)(cid:1) = c n lim Λ →∞ ˆ B (0 , Λ) ∩ T [ B (0 , Λ)] M U ( x ) d n x, proving the assertion. (cid:3) Finally, we apply Theorem 10.9 and prove that for admissible potentials Φ,ind( Q + Φ) = 0: Corollary 10.11.
Let n ∈ N > odd, d ∈ N . Let Φ be admissible, see Definition . Let Q be as in (6.3) and L = Q + Φ as in (7.1) . Then L is Fredholm and ind( L ) = 0 .Proof. By invariance of the Fredholm index under relatively compact perturbations(cf. Theorem 3.6 ( iii )), we can assume without loss of generality, that Φ is constantin a neighborhood of 0. We consider T : R n \{ } → R n given by T ( x ) := x | x | , x ∈ R n \{ } . One observes that T is a transformation of constant orientation. Moreover, as Φ isadmissible, so is Φ T := Φ ◦ T . In particular, since Φ is constant in a neighborhoodof 0, we find Λ > x ∈ R n with | x | > Λ, ( ∂ i Φ T )( x ) = 0. Hence,ind( Q + Φ T ) = 0, by Theorem 10.1 and, thus ind( Q + Φ) = 0, by Theorem 10.9. (cid:3) For an entirely diffferent approach to Corollary 10.11 we refer again to [86,Theorem 5.2].
Remark . As kindly pointed out to us by one of the referees, Corollary 10.11permits a more elementary proof as follows. If Φ is admissible, then x Φ t ( x ) :=Φ( tx ), t >
0, is also admissible and the associated operators L t : H ( R n ) b n d → L ( R n ) b n d , t >
0, are all Fredholm. In addition, the map, (0 , ∞ ) ∋ t L t ∈B (cid:0) H ( R n ) b n d , L ( R n ) b n d (cid:1) is continuous. Thus (cf. Corollary 3.7),ind( L ) = ind( L t ) , t > . (10.5)However, (6.14) leads to L ∗ t L t = − ∆ I b n d − C t + Φ t , L t L ∗ t = − ∆ I b n d + C t + Φ t , (10.6)where C t = n X j =1 γ j,n ( ∂ j Φ t ) = ( Q Φ t ) , t > . (10.7)Hence, for some constant c > k C t k B ( L ( R n ) b nd ) c t for 0 < t sufficiently small. Inparticular, for 0 < t sufficiently small, the operators L ∗ t L t and L t L ∗ t are boundedlyinvertible and hence ind( L t ) = 0, implying ind( L ) = ind( L ) = 0. ⋄ Perturbation Theory for the Helmholtz Equation
Before we are in a position to provide a proof of Theorem 10.2, we need someresults concerning the perturbation theory of Helmholtz operators. More precisely,we study operators (and their fundamental solutions) of the form( − ∆ + µ + η )in odd space dimensions n > η ∈ L ∞ ( R n ) with small support around theorigin and µ ∈ C Re > . For µ ∈ C Re > , η ∈ L ∞ ( R n ), recalling R µ = ( − ∆ + µ ) − ,one formally computes R η + µ := (cid:0) − ∆ + η + µ ) (cid:1) − = (cid:0) ( − ∆ + µ )(1 + R µ η (cid:1) − = ∞ X k =0 ( R µ ( − η )) k R µ . (11.1)This computation can be made rigorous, if k R µ ( η ) k B ( L ( R n )) <
1. The first aimof this section is to provide a proof of the fact that if k η k L ∞
1, then indeed k R µ ( η ) k B ( L ( R n )) < µ , that is, for µ belonging to theclosed sector Σ µ ,ϑ = { z ∈ C | Re( µ ) > µ , | arg( µ ) | ϑ } (11.2)for some µ ∈ R , ϑ ∈ [0 , π ], provided the support of η is sufficiently small.For µ > x, y ∈ R n , x = y , we introduce s µ ( x − y ) := e −√ µ | x − y | | x − y | n − . (11.3)The next lemma shows that the Helmholtz Green’s function basically behaveslike s µ in (11.3). We note that a similar estimate was used in [22, p. 224, formula(c)]. However, we further remark that the factor λ introduced in the followingresult does not occur in [22, p. 224, formula (c)], yielding a hidden z -dependenceof the constant K occuring there. Lemma 11.1.
Let n ∈ N > odd, λ ∈ (0 , . For µ > denote the integral kernelof R µ = ( − ∆ + µ ) − in L ( R n ) by r µ , see Lemma or (5.11) , and let s µ be asin (11.3) . Then there exist c , c > such that for all µ > , r µ ( x − y ) c s λµ ( x − y ) , and s µ ( x − y ) c r µ ( x − y ) , x, y ∈ R n , x = y. Proof.
For the first inequality, one observes that for k ∈ { , . . . , b n − } , with n =2 b n + 1, the function R > ∋ β β k e − (1 −√ λ ) β is bounded by some d k >
0. Next, let x, y ∈ R n , x = y and r := | x − y | , µ > k ∈ { , . . . , b n − } , one estimates e −√ µ | x − y | | x − y | n − − k ( √ µ ) k = e −√ µr r n − ( r √ µ ) k d k e −√ λµr r n − . Hence, the first inequality asserted follows from Lemma 5.11. Employing againLemma 5.11, the second inequality can be derived easily. (cid:3)
We can now come to the announced result of bounding the operator norm of R µ η given η is supported on a small set. We note that smallness of the support isindependent of µ , if one assumes µ to lie in a sector. HE CALLIAS INDEX FORMULA REVISITED 87
Lemma 11.2.
Let µ > , ϑ ∈ (0 , π/ , β > , n ∈ N > odd. Then there exists τ > such that for all µ ∈ Σ µ ,ϑ , see (11.2) , k R µ η k B ( L ( R n )) β for all η ∈ L ∞ ( R n ) , k η k L ∞ and supp( η ) ⊂ B (0 , τ ) .Proof. Let τ > η ∈ L ∞ ( R n ) such that supp( η ) ⊂ B (0 , τ ) and k η k L ∞ µ ∈ Σ µ ,ϑ and denote the fundamental solution of ( − ∆ + µ ) by r µ , see alsoLemma 5.12. By estimate (5.15) in Lemma 5.12, there exists c > | r µ ( x − y ) | c r Re µ ( x − y ) , x, y ∈ R n , x = y, µ ∈ Σ µ ,ϑ . Next, by Lemma 11.1, there exists c > µ > µ , r µ ( x − y ) c s µ ( x − y ) , x, y ∈ R n , x = y. For µ > µ , one notes that k s µ/ k L ( R n ) k s µ / k L ( R n ) < ∞ . Hence, for µ ∈ Σ µ ,ϑ and u ∈ C ∞ ( R n ), one gets k R µ ( η ) u k L ( R n ) = ˆ R n (cid:12)(cid:12)(cid:12)(cid:12) ˆ R n r µ ( x − y ) η ( y ) u ( y ) d n y (cid:12)(cid:12)(cid:12)(cid:12) d n x = ˆ R n (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (0 ,τ ) r µ ( x − y ) η ( y ) u ( y ) d n y (cid:12)(cid:12)(cid:12)(cid:12) d n x c ˆ R n (cid:18) ˆ B (0 ,τ ) r Re( µ ) ( x − y ) | η ( y ) || u ( y ) | d n y (cid:19) d n x c c ˆ R n (cid:18) ˆ B (0 ,τ ) s Re( µ ) ( x − y ) d n y (cid:19) ˆ B (0 ,τ ) s Re( µ ) ( x − y ) | u ( y ) | d n yd n x c c ˆ R n (cid:18) ˆ B (0 ,τ ) s Re( µ ) ( y ) d n y (cid:19) ˆ R n s Re( µ ) ( x − y ) | u ( y ) | d n yd n x = c c ˆ B (0 ,τ ) s Re( µ ) ( y ) d n y k s µ k L ( R n ) k u k L ( R n ) . One observes that ˆ B (0 ,τ ) s Re( µ ) ( y ) d n y ω n − ˆ τ r n − r n − dr = τ ω n − , and hence, k R µ η k B ( L ( R n )) c c s k s µ / k L ( R n ) ω n − τ. (cid:3) Remark . ( i ) Let µ >
0, and ϑ ∈ (0 , π ), κ >
0. Then for all µ ∈ Σ µ ,ϑ (see(11.2)), there exists τ > η ∈ L ∞ ( R n ), with k η k L ∞ κ and η = 0on R n \ B (0 , τ ), the operator R η + µ = ( − ∆ + η + µ ) − exists as a bounded linearoperator in L ( R n ) and its norm is arbitarily close to k R µ k . Indeed, for β < k R µ η k β one computes k R η + µ k ∞ X k =0 β k k R µ k = 11 − β k R µ k . ( ii ) In the situation of part ( i ), we shall now elaborate some more on the propertiesof R η + µ with k R µ η k β < µ ∈ Σ µ ,ϑ . Assuming, in addition, η ∈ C ∞ , then R η + µ extends by interpolation to a bounded linear operator to the full Sobolevscale H s ( R n ) (see (5.1) for a definition), s ∈ R . Moreover, from( − ∆ + µ ) R η + µ = ( − ∆ + µ ) ∞ X k =0 R µ ( − ηR µ ) k = ∞ X k =0 (( − η ) R µ ) k , one gets k ( − ∆ + µ ) R η + µ k (1 − β ) − , yielding R η + µ ∈ B ( H s ( R n ); H s +2 ( R n )) forall s ∈ R . ⋄ With Lemma 11.2 we have an a priori condition on the support of η to makethe operator R η + µ well-defined. The forthcoming results, the very reason of thisentire section, provide estimates for the integral kernels of the perturbed operatorin terms of the unperturbed one. Of course, these estimates also rely on a Neumannseries type argument. The main step is the following lemma. Lemma 11.4.
Let n ∈ N > odd, µ > , κ > . For any λ ∈ (0 , , there exists τ > such that for all η ∈ L ∞ ( R n ) , with k η k L ∞ κ and supp( η ) ⊂ B (0 , τ ) , suchthat for all k ∈ N > , µ > µ , the integral kernel e r k of ( R µ η ) k R µ satisfies | e r k ( x, y ) | λ k r µ/ ( x − y ) , x, y ∈ R n , x = y, where r µ is the integral kernel of R µ = ( − ∆ + µ ) − given by (5.11) . We postpone the proof of Lemma 11.4 and show three preparatory results first.
Lemma 11.5.
Let n ∈ N > odd, µ > , τ > , s µ as in (11.3) . Then for all x, z ∈ R n , x = z , the inequality, ˆ B (0 ,τ ) s µ ( x − y ) s µ ( y − z ) d n y n − ω n − τ s µ ( x − z ) , holds, with ω n − the ( n − -dimensional volume of the unit sphere S n − ⊆ R n ( seealso (5.6)) .Proof. One notes that, by the triangle inequality, e −√ µ | x − y | e −√ µ | y − z | e −√ µ | x − z | , x, y, z ∈ R n . Hence, one is left with showing ˆ B (0 ,τ ) | x − y | n − | y − z | n − d n y n − ω n − τ | x − z | n − , x, y, z ∈ R n , x = z. Let x, y, z ∈ R n . Then | x − z | n − ( | x − y | + | y − z | ) n − n − ( | x − y | n − + | y − z | n − ) . Hence, ˆ B (0 ,τ ) | x − z | n − | x − y | n − | y − z | n − d n y ˆ B (0 ,τ ) n − ( | x − y | n − + | y − z | n − ) | x − y | n − | y − z | n − d n y = 2 n − ˆ B (0 ,τ ) (cid:18) | x − y | n − + 1 | y − z | n − (cid:19) d n y n − ˆ B (0 ,τ ) | y | n − d n y HE CALLIAS INDEX FORMULA REVISITED 89 = 2 n − ω n − ˆ τ r dr = 2 n − ω n − τ . (cid:3) Proposition 11.6.
Let n ∈ N , µ > , and q ∈ L ( R n ) ∩ C ( R n \{ } ) . Assume that V q , the operator defined by convolution with q , defines a self-adjoint, nonnegativeoperator in L ( R n ) . Then for all µ > µ and x, y ∈ R n , x = y , µ ˆ R n r µ ( x − x ) q ( x − y ) d n x q ( x − y ) . Proof.
Let µ > µ . As the convolution with q commutes with differentiation, italso commutes with ( − ∆ + µ ), R µ or powers thereof. Since V q >
0, there existsa unique nonnegative square root V / q , which also commutes with R µ , R − µ andpowers thereof. For φ ∈ H ( R n ) and µ > µ , (cid:0) ( − ∆ + µ ) φ, V q φ (cid:1) L = (cid:0) ( − ∆ + µ ) V / q φ, V / q φ (cid:1) L > µ (cid:0) V / q φ, V / q φ (cid:1) L = µ (cid:0) φ, V q φ (cid:1) L . Putting φ := ( − ∆ + µ ) − / ψ = R / µ ψ for some ψ ∈ H ( R n ), one infers (cid:0) ψ, V q ψ (cid:1) L > µ (cid:0) R / µ ψ, V q R / µ ψ (cid:1) L > µ (cid:0) ψ, R µ V q ψ (cid:1) L . As ( − ∆ + µ ) − / [ H ( R n )] is dense in L ( R n ), it follows that V q − µ R µ V q is anonnegative integral operator, which implies the asserted inequality. (cid:3) Applying Proposition 11.6 with q = r µ twice, one gets the proof of the followingresult. Lemma 11.7.
Let n ∈ N , µ > . Then for all µ > µ , µ ˆ ( R n ) r µ ( x − x ) r µ ( x − x ) r µ ( x − y ) d n x d n x r µ ( x − y ) , x, y ∈ R n , x = y. (11.4) Proof of Lemma . Recalling (11.3), s µ ( x − y ) = e −√ µ | x − y | | x − y | n − , x, y ∈ R n , x = y, µ > . By Lemma 11.1, there exist c , c > µ > x, y ∈ R n , x = y , r µ ( x − y ) c s µ/ ( x − y ) , s µ/ ( x − y ) c r µ/ ( x − y ) . (11.5)Next, one recalls, with n = 2 b n + 1 for some b n ∈ N , from Lemma 5.12, equation(5.16), that r µ ( x − y ) b n − r µ/ ( x − y ) , x, y ∈ R n , x = y. Let τ >
0. We estimate for x, y ∈ R n , x = y , η ∈ L ∞ ( R n ), with supp( η ) ⊂ B (0 , τ ), µ > µ , using Lemma 11.5 and inequality (11.5), with κ τ := 2 n − ω n − τ , | e r k ( x, y ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( R n ) k r µ ( x − x ) η ( x ) r µ ( x − x ) · · · η ( x k ) r µ ( x k − y ) d n x · · · d n x k (cid:12)(cid:12)(cid:12)(cid:12) k η k kL ∞ ˆ ( B (0 ,τ )) k r µ ( x − x ) r µ ( x − x ) · · · r µ ( x k − y ) d n x · · · d n x k k η k kL ∞ c k − ˆ ( B (0 ,τ )) k r µ ( x − x ) s µ/ ( x − x ) · · · × s µ/ ( x k − − x k ) r µ ( x k − y ) d n x · · · d n x k k η k kL ∞ ( c κ τ ) k − ˆ ( B (0 ,τ )) r µ ( x − x ) s µ/ ( x − x k ) r µ ( x k − y ) d n x d n x k k η k kL ∞ ( c κ τ ) k − c ˆ ( B (0 ,τ )) r µ ( x − x ) r µ/ ( x − x k ) r µ ( x k − y ) d n x d n x k k η k kL ∞ ( c κ τ ) k − n − c ˆ ( R n ) r µ/ ( x − x ) r µ/ ( x − x k ) × r µ/ ( x k − y ) d n x d n x k k η k kL ∞ µ ( c κ τ ) k − n − c r µ/ ( x − y ) , where, in the last estimate, we used Lemma 11.7. (cid:3) Having proved Lemma 11.4, we can now formulate and prove the result for theestimate of the perturbed and the unperturbed integral kernels (Green’s functions).
Theorem 11.8.
Let n ∈ N > odd, µ > , ϑ ∈ (0 , π/ , κ > . Then there exists c, τ > such that for all µ ∈ Σ µ ,ϑ and η ∈ C ∞ ( R n ) with supp( η ) ⊂ B (0 , τ ) , k η k L ∞ κ , the estimate | r η + µ ( x, y ) | cr Re( µ ) ( x − y ) , x, y ∈ R n , x = y, holds, where r η + µ and r Re( µ ) are the integral kernels for the operators R η + µ =( − ∆ + η + µ ) − and R Re( µ ) , respectively.Proof. One recalls that r µ denotes the integral kernel of R µ . According to Lemma5.12, (5.15), there exists c > µ ∈ Σ µ ,ϑ one has | r µ ( x − y ) | c r Re( µ ) ( x − y ) , x, y ∈ R n , x = y. Next, by Lemma 11.2, one chooses τ > k R µ η k / µ ∈ Σ µ ,ϑ and η ∈ L ∞ ( R n ) with supp( η ) ⊂ B (0 , τ ) and k η k L ∞ κ , implying that R η + µ is awell-defined bounded linear operator in L ( R n ) (see, e.g., Remark 11.3).Let τ > k ∈ N > , the integral kernel e r k, Re( µ ) for theoperator ( R Re( µ ) η ) k R Re( µ ) satisfies | e r k, Re( µ ) ( x, y ) | c c ) k r Re( µ ) / ( x − y ) (11.6)for all x, y ∈ R n , x = y , µ ∈ Σ µ ,ϑ , η ∈ L ∞ ( R n ) with k η k L ∞ κ and supp( η ) ⊂ B (0 , τ ) and some c >
0, which is possible by Lemma 11.4.Let τ := min { τ , τ } . Then, for x, y ∈ R n , and η ∈ C ∞ ( R n ), with supp( η ) ⊂ B (0 , τ ) and k η k L ∞ κ one gets for N, M ∈ N , N > M , µ ∈ Σ µ ,ϑ , (cid:12)(cid:12)(cid:12)(cid:12) N X k = M (cid:0) ( r µ ∗ η ) · · · ( r µ ∗ η ) | {z } k -times (cid:1) r µ ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12) N X k = M c k +11 | e r k, Re( µ ) ( x, y ) | c c ∞ X k = M − k r Re( µ ) / ( x − y ) (11.7) c c − M +1 r Re( µ ) / ( x − y ) . HE CALLIAS INDEX FORMULA REVISITED 91
Thus, e r ( x, y ) := ∞ X k =0 (cid:0) ( r µ ∗ η ) · · · ( r µ ∗ η ) | {z } k -times (cid:1) r µ ( x − y ) , x, y ∈ R n , x = y, defines a function, which, by the differentiability of η , coincides with the funda-mental solution of ( − ∆ + η + µ ). For x, y ∈ R n , x = y , µ ∈ Σ µ ,ϑ , one thus gets,using (11.7) for M = 1 and N → ∞ , | r η + µ ( x, y ) | | r η + µ ( x, y ) − r µ ( x − y ) | + | r µ ( x − y ) | c c r Re( µ ) / ( x − y ) + c r Re( µ ) ( x − y ) ( c c + c b n − ) r Re( µ ) / ( x − y ) , where, in the last estimate, we used Lemma 5.12, (5.16), with n = 2 b n + 1 for some b n ∈ N . Finally, for µ ∈ Σ µ ,ϑ , and from R η + µ = ∞ X k =0 R µ (( − η ) R µ ) k = R µ − R µ ηR η + µ , one reads off, for x, y ∈ R n , x = y , | r η + µ ( x, y ) | | r µ ( x − y ) | + | r µ ∗ ηr η + µ ( x, y ) | c r Re( µ ) ( x − y ) + c κ ( c c + c b n − ) r Re( µ ) ∗ r Re( µ ) / ( x − y ) c (cid:18) κ ( c c + c b n − ) 4 µ (cid:19) r Re( µ ) ( x − y ) , where we used Proposition 11.6 for q = r Re( µ ) / for obtaining the last estimate. (cid:3) As a first application of Theorem 11.8, in the spirit of the results derived inSection 5, we can show the following result.
Corollary 11.9.
Let n ∈ N > odd, with n = 2 b n + 1 for some b n ∈ N , µ > , ϑ ∈ (0 , π/ . Then there exists τ > , such that for m ∈ N > b n there exists c > with the following properties: Given Ψ , . . . , Ψ m ∈ C ∞ b ( R n ) , with | Ψ j ( x ) | κ (1 + | x | ) − α j , x ∈ R n , j ∈ { , . . . , m } , for some α , . . . , α m , κ ∈ [0 , ∞ ) , then for all η j ∈ C ∞ b ( R n ) , k η j k L ∞ , j ∈{ , . . . , m } , and supp( η j ) ⊂ B (0 , τ ) , the integral kernel t µ of Q j ∈{ ,...,m } R η j + µ Ψ j satisfies | t µ ( x, x ) | κ m c (1 + | x | ) − P mj =1 α j , x ∈ R n , µ ∈ Σ µ ,ϑ . Proof.
Choose τ > τ ’s according to Theorem 11.8 with κ = 1and Lemma 11.2 with β = . Let Ψ , . . . , Ψ m , η , . . . , η m , m as in Corollary 11.9,and let κ ′ > κ . Choose e Ψ j ∈ C ∞ b ( R n ; [0 , ∞ )) with | Ψ j ( x ) | e Ψ j ( x ) κ ′ (1 + | x | ) − α j , x ∈ R n , j ∈ { , . . . , m } . Then, by Theorem 11.8, there exists c > | r η j + µ ( x, y ) | cr Re( µ ) ( x − y ) , x, y ∈ R n , x = y, µ ∈ Σ µ ,ϑ . Hence, for x ∈ R n and µ ∈ Σ µ ,ϑ one obtains | (( r η + µ ∗ Ψ ) · · · ( r η m + µ ∗ Ψ m ))( x, x ) | c m (cid:0) ( r Re( µ ) ∗ e Ψ ) · · · ( r Re( µ ) ∗ e Ψ m ) (cid:1) ( x, x ) . Thus, the assertion follows from Lemma 5.14. (cid:3)
Remark . A result similar to Corollary 11.9 holds if for some index j ∈{ , . . . , m } , the operator R η j + µ is replaced by ∂ ℓ R η j + µ for some ℓ ∈ { , . . . , n } .For obtaining such a result, one needs a version of Lemma 5.13 where, in thislemma, the fundamental solution for the Helmholtz equation is replaced by therespective one for ( − ∆ + η j + µ ) u = f . ⋄ In the rest of this section, we shall establish the remaining estimate needed, toobtain a proof for Remark 11.10. More precisely, we aim for a proof of the followingresult:
Theorem 11.11.
Let n ∈ N > odd, for µ ∈ C Re > , let q µ as in Lemma , µ > , ϑ ∈ (0 , π/ , κ > . Then there exists c > and τ > such that for all j ∈ { , . . . , n } , η ∈ C ∞ b ( R n ) , k η k L ∞ κ , with supp( η ) ⊂ B (0 , τ ) , and µ ∈ Σ µ ,ϑ ,we have for all x, y ∈ R n , x = y , | ∂ j ( ξ r η + µ ( ξ, y ))( x ) | c q Re( µ ) ( | x − y | ) with r η + µ denoting the integral kernel of R η + µ = ( − ∆ + η + µ ) − , the latter beinggiven by (11.1) . The proof of Theorem 11.11 will follow similar ideas as the one for Theorem11.8. We start with the following result:
Theorem 11.12.
Let n ∈ N > , k ∈ N , k < n , µ > . Then the operator L ( R n ) ∋ ψ (cid:18) x ˆ R n e − µ | x − y | | x − y | k ψ ( y ) d n y (cid:19) ∈ L ( R n ) is well-defined, bounded, and positive definite.Proof. The operator is well-defined and bounded by Young’s inequality togetherwith the observation that f : x e − µ | x | | x | − k is an L ( R n )-function. Moreover, for ε > φ ε : [0 , ∞ ) → R , r e − µr ( r + ε ) k . Then φ ε is a completely monotone function, since the maps r e − µr and r ( r + ε ) − k are completely monotone. Observing that φ ε ( r ) → r → ∞ and usingthe criterion on positive definiteness in [97, Theorem 2] one infers that φ ε ( | · | ) ∗ isa positive semi-definite operator. Moreover, since φ ε → φ in L ( R n ) as ε →
0, onegets that φ ε ( | · | ) ∗ → φ ( | · | ) ∗ in B ( L ( R n )) as ε →
0. Hence, for all ψ ∈ L ( R n ) oneinfers 0 lim ε → (cid:0) φ ε ∗ ψ, ψ (cid:1) L ( R n ) = (cid:0) φ ∗ ψ, ψ (cid:1) L ( R n ) . (cid:3) Corollary 11.13.
Let n ∈ N > , k ∈ N , k < n , µ , µ > . Denote q : R n \{ } ∋ x e − µ | x | | x | − k . Then for all µ > µ , µ ˆ R n r µ ( x − x ) q ( x − x ) d n x q ( x − y ) , x, y ∈ R n , x = y, (11.8) where r µ is the integral kernel for ( − ∆ + µ ) − ∈ B ( L ( R n )) .Proof. By Theorem 11.12, q satisfies the assumptions in Proposition 11.6, implyinginequality (11.8). (cid:3) We conclude with the proof of Theorem 11.11, yielding the proof of Remark11.10.
HE CALLIAS INDEX FORMULA REVISITED 93
Proof of Theorem . Choose τ > k R µ η k / η ∈ L ∞ ( R n ), k η k L ∞ κ , with supp( η ) ⊂ B (0 , τ ), and µ ∈ Σ µ ,ϑ , as permitted by Lemma 11.2.Next, let j ∈ { , . . . , n } and recall ∂ j R η + µ = ∂ j ∞ X k =0 R µ (cid:0) ( − η ) R µ (cid:1) k = ∂ j R µ + ∂ j R µ ( − η ) R µ ∞ X k =1 (cid:0) ( − η ) R µ (cid:1) k − = ∂ j R µ + ∂ j R µ ( − η ) ∞ X k =0 R µ (cid:0) ( − η ) R µ (cid:1) k = ∂ j R µ − ∂ j R µ ( η ) R η + µ . (11.9)Let q µ be as in Lemma 5.13. Upon appealing to Lemma 5.13 (see, in particular,inequalities (5.18) and (5.19)), one is left with estimating the integral kernel asso-ciated with the second summand in (11.9), which we denote by t . Using Theorem11.8 and Lemma 5.13, (5.19), there exists c > | r η + µ ( x, y ) | c r Re( µ ) / ( x − y ) and q µ ( | x − y | ) c q Re( µ ) ( | x − y | )for all µ ∈ Σ µ ,ϑ and x, y ∈ R n , x = y . Thus, for all x, y ∈ R n , x = y , µ ∈ Σ µ ,ϑ ,one gets with the help of (11.8) (using that q Re( µ ) ( | · | ) is a nonnegative linearcombination of functions discussed in Corollary 11.13), | t ( x, y ) | = (cid:12)(cid:12) ∂ j r µ ∗ ( η ) r η + µ ( x, y ) (cid:12)(cid:12) c ˆ B (0 ,τ ) q µ ( | x − x | ) | η ( x ) | r Re( µ ) / ( x − y ) d n x k η k L ∞ c ˆ R n q Re( µ ) ( | x − x | ) r Re( µ ) / ( x − y ) d n x k η k L ∞ c µ q Re( µ ) ( | x − y | ) . (cid:3) The proof of Theorem : The Smooth Case
In this section, we treat Theorem 10.2 for the particular case of C ∞ -potentials .Let n ∈ N > odd, Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) for some d ∈ N . Assume that Φ( x ) = Φ( x ) ∗ and that for some c > R > | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ). With the operator L = Q + Φ as in (7.1),we proceed as follows: At first, we show that if ( Q Φ)( x ) → | x | → ∞ , then L is a Fredholm operator (Lemma 12.1). Next, if one defines U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) tocoincide with sgn(Φ) on R n \ B (0 , R ) as in Lemma 10.4, we show that the operator Q + U is also a Fredholm operator with the same index (Theorem 12.2). Moreover,in this theorem, we shall also show that changing U to be unitary everywhere buton a small ball around 0 will not change the index. As this ball may be chosenarbitrarily small, we are in the position to proceed with a similar strategy to derivethe index as in Section 7 and use the results from Section 11. In that sense, thefollowing may also be considered as a first attempt for a perturbation theory forthe generalized Witten index introduced at the end of Theorem 3.4.We start with the Fredholm property for the operator considered in Theorem10.2 with smooth potentials (see also Theorem 6.3). Lemma 12.1.
Let n, d ∈ N , L = Q + Φ as in (7.1) , with Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) and Φ( x ) = Φ( x ) ∗ , x ∈ R n . Assume that C ( x ) := ( Q Φ) ( x ) → as | x | → ∞ ( see also (6.15)) , and that there exist c > and R > such that with | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ) . Then L is a Fredholm operator.Proof. One recalls from Proposition 6.10 that L ∗ L = − ∆ − C + Φ and LL ∗ = − ∆ + C + Φ . The latter two operators are ∆-compact perturbations of − ∆ + Φ due to C ( x ) → | x | → ∞ and Theorem 6.7. Next, since − ∆ + Φ + c χ B (0 ,R ) > − ∆+ c , the operator − ∆+Φ + c χ B (0 ,R ) is continuously invertible. But, − ∆+Φ + c χ B (0 ,R ) is also a ∆-compact perturbation of − ∆ + Φ . Thus, by the invariance ofthe Fredholm property under relatively compact perturbations, one concludes theFredholm property for − ∆ + Φ and thus the same for L ∗ L and LL ∗ . (cid:3) As a corollary, we obtain the assertion that one might also consider potentialsbeing pointwise unitary outside large balls. In this context, we refer the reader alsoto the beginning of Section 10. One notes that also Theorem 10.2 hints in the samedirection as in the index formula only the sign of the potential occurs.
Theorem 12.2.
Let n, d ∈ N , L = Q + Φ as in (7.1) with Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) with Φ( x ) = Φ( x ) ∗ , x ∈ R n . Assume that there exist c > and R > suchthat | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ) and ( Q Φ)( x ) → as | x | → ∞ . If U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) pointwise self-adjoint with sgn(Φ( x )) = U ( x ) for all x ∈ R n with | x | > R ′ for some R ′ > R , then e L := Q + U is Fredholm and ind( L ) = ind (cid:0)e L (cid:1) .Proof. From Lemma 10.4, one gets ( Q U )( x ) = ( Q sgn(Φ))( x ) → | x | → ∞ .Hence, by Lemma 12.1 the operators L = Q + Φ and e L = Q + U are Fredholm(for e L one observes that U ( x ) = I d for all | x | > R ′ ). Next, the operator family[0 , ∋ λ
7→ Q + (1 − λ ) U + λ min { c/ , / } U defines a homotopy from e L to Q + min { c/ , / } U , which is a homotopy of Fredholm operators as (1 − λ ) + λ min { c/ , / } = 1 − λ (1 − min { c/ , / } ) > min { c/ , / } > λ ∈ (0 , We note that this section may explain the reasoning underlying the last lines on [22, p. 226].
HE CALLIAS INDEX FORMULA REVISITED 95 that [0 , ∋ λ
7→ Q + (1 − λ )Φ + λ min { c/ , / } U defines a homotopy of Fredholmoperators. Employing Lemma 12.1, it suffices to show that for some e c > (cid:2) (1 − λ )Φ( x ) + λ min { c/ , / } U ( x ) (cid:3) > e cI d for all x ∈ R n \ B (0 , R ′ ) and λ ∈ [0 , d × d -matrices it suffices to show that for real numbers α ∈ R with α > c , onehas for some e c > (cid:2) (1 − λ ) α + λ min { c/ , / } sgn( α ) (cid:3) > e c. But since (cid:2) (1 − λ ) α + λ min { c/ , / } sgn( α ) (cid:3) = (cid:2) (1 − λ ) | α | + λ min { c/ , / } (cid:3) , it remains to observe that(1 − λ ) | α | + λ min { c/ , / } > (1 − λ ) c + λ min { c/ , / } > min { c/ , / } . (cid:3) We remark that the assumptions in Theorem 12.2 can be met, using Lemma10.4. This, and [22] motivates the following notion of “Callias admissibility”.
Definition 12.3.
Let n, d ∈ N . We say that a map Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is called Callias admissible , if the following conditions ( i ) – ( iii ) are satisfied: ( i ) Φ( x ) = Φ( x ) ∗ , x ∈ R n . ( ii ) There exists
R > such that Φ( x ) is unitary for all x ∈ R n \ B (0 , R ) . ( iii ) There exists ε > / such that for all α ∈ N n , there is κ > with k ∂ α Φ( x ) k κ ( (1 + | x | ) − , | α | = 1 , (1 + | x | ) − − ε , | α | > , x ∈ R n . Remark . Let Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be Callias admissible. By Theorem 12.2,for any potential U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) coinciding with sgn(Φ) on large balls, theoperators L = Q + Φ and e L = Q + U are Fredholm with the same index. Moreover,by Theorem 10.3 and the unitarity of Φ on large balls, one infers that on large balls U = Φ. Next, by Lemma 10.4, we may choose U to be unitary everywhere but ona small ball centered at 0. In addition, we can choose U such that U ( x ) = u ( x ) I d , x ∈ R n , with u ∈ C ∞ ( R n ; [0 , u = 1 on R n \ B (0 , τ ) for every chosen τ >
0. For thatreason, in order to compute the index for L = Q + Φ, our main focus only needsto be potentials with the properties of U discussed here, and then one can employthe results of Section 11. ⋄ Remark 12.4 leads to the following definition:
Definition 12.5 ( τ -admissibility) . Let n, d ∈ R n , τ > , U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) .We say that U is admissible on R n \ B (0 , τ ) , in short, τ -admissible , if U is Calliasadmissible and there exists u ∈ C ∞ ( R n ; [0 , satisfying U ( x ) = u ( x ) I d , x ∈ R n , (12.1) with the property that u = 1 on R n \ B (0 , τ ) . To get a first impression of the difference between the notions of admissibility(see Definition 6.11) and τ -admissibility, we will compute the resolvent differencesof the resolvents of L ∗ L and LL ∗ in Proposition 12.7, with L = Q + U given as in(7.1) for some τ -admissible U with u as in (12.4). First, we note that by Proposition6.10, one has L ∗ L = − ∆ I b n d + C + uI b n d , with C = ( Q U ). In Section 7, and in particular in Proposition 7.5, we discussedthe resolvent ( L ∗ L + z ) − in terms of R z = ( − ∆ I b n d + (1 + z )) − . The latteroperator needs to be replaced by the following (see also (11.1)) R u + z := ( − ∆ I b n d + u + z ) − (12.2)= ( − ∆ I b n d + ( u − I b n d + ( z + 1) I b n d ) − = ∞ X k =0 ( R z ( u − I b n d ) k R z , provided the latter series converges. As already discussed in Lemma 10.4, this canbe ensured if τ is chosen small enough. Thus, for this pupose, we shall fix theparameters according to the results in Section 11: Hypothesis 12.6.
Let n = 2 b n + 1 , b n ∈ N > , δ ∈ ( − , , ϑ ∈ (0 , π/ . (12.3) For µ := δ + 1 let τ as in Lemma for β = 1 / , τ as in Theorem for κ = 1 , τ as in Theorem for κ = 1 , and τ as in Corollary . Define τ := min { τ , τ , τ , τ } . (12.4)As mentioned already, for τ -admissible potentials, we shall derive the indextheorem similarly to the derivation for admissible potentials. More precisely, atfirst, we will focus on computing the trace of χ Λ B L ( z ), as in Theorem 7.1. We notethat the following parallels the Section 7.To start, we need to state a result similar to Proposition 7.5. In fact, using theexpressions in (7.11) and (7.10), with R z replaced by R u + z (see (11.1)), even theproof turns out to be the same. Proposition 12.7.
Assume Hypothesis , let z ∈ Σ δ ,ϑ , and suppose U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is τ -admissible ( cf. Definition , with u as in (12.1)) . We recallthat L = Q + U as in (7.1) , C = ( Q U ) in (6.15) , and R u + z in (12.2) . If, inaddition, z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) , then for all N ∈ N , ( L ∗ L + z ) − − ( LL ∗ + z ) − = 2 N X k =0 R u + z ( CR u + z ) k +1 + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR u + z ) N +2 = 2 N X k =0 R u + z ( CR u + z ) k +1 + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) N +3 , and ( L ∗ L + z ) − + ( LL ∗ + z ) − = 2 N X k =0 R u + z ( CR u + z ) k + (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) N +2 . Next, we formulate the variant of Lemma 7.7:
Lemma 12.8.
Assume Hypothesis , z ∈ Σ δ ,ϑ , let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be τ -admissible ( cf. Definition , with u as in (12.1) . Let L = Q + U be given by (7.1) and z ∈ Σ δ ,ϑ ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) . We recall B L ( z ) , J jL ( z ) , and A L ( z ) given by (7.2) , (7.6) , and (7.7) ( with Φ replaced by U ) , respectively, as well as R u + z given by (12.2) . Then the following assertions hold: B L ( z ) = n X j =1 (cid:2) ∂ j , J jL ( z ) (cid:3) + A L ( z ) , = z tr b n d (cid:0) R u + z C ) n R u + z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR u + z ) n +1 (cid:1) , with J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R u + z C ) n − R u + z (cid:1) + 2 tr b n d (cid:0) γ j,n U ( R u + z C ) n − R u + z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) + tr b n d (cid:0) γ j,n U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) ,j ∈ { , . . . , n } , and A L ( z ) = tr b n d (cid:0)(cid:2) U, U (cid:0) R u + z C ) n R u + z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR u + z ) n +1 (cid:1)(cid:3)(cid:1) − tr b n d (cid:0)(cid:2) U, Q (cid:0) R u + z C ) n − R u + z + (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) × ( CR u + z ) n (cid:1)(cid:3)(cid:1) . Proof.
The proof follows line by line those of Lemma 7.7, observing that R u + z commutes with γ j,n , j ∈ { , . . . , n } . (cid:3) Remark . For even space dimensions n – as in Lemma 7.7 – the correspondingoperator B L ( z ) also vanishes for all z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ). That is why we willdisregard even space dimensions from now on. ⋄ The proof of the variant of Theorem 7.8 is slightly more involved:
Theorem 12.10.
Assume Hypothesis , z ∈ Σ δ ,ϑ . Let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be τ -admissible ( cf. Definition , with u as in (12.1) . Let L = Q + U be given by (7.1) . Then there exists δ δ < , such that for all z ∈ Σ δ,ϑ ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) and Λ > , the operator χ Λ B L ( z ) , with B L ( z ) given by (7.2) , is trace class with z tr( | χ Λ B L ( z ) | ) bounded on B (0 , | δ | ) \{ } . Moreover, the trace of χ Λ B L ( z ) maybe computed as the integral over the diagonal of the corresponding integral kernel.Proof. It suffices to observe that if η ∈ L n +1 ( R n ), one has R u + z η ∈ B n +1 ( L ( R n )),with k R u + z η k B n +1 k R z η k B n +1 . Indeed, from R u + z η = ∞ X k =0 ( R z ( u − k R z η, the ideal property, Hypothesis 12.6, and (11.1), it follows that k R u + z η k B n +1 = ∞ X k =0 k ( R z ( u − k k B ∞ k R z η k B n +1 k R z η k B n +1 . The rest of the proof of the trace class property follows literally that of Theorem7.8. The assertion concerning the computation of the trace rests on Remark 7.9,which applies in this context. (cid:3)
The variant of Lemma 8.1 with L = Q + U instead of L = Q + Φ for some τ -admissible U need not be stated again as it only contains a statement aboutthe regularity of the integral kernels of J jL ( z ) and A L ( z ) (see Lemma 12.8), j ∈{ , . . . , n } . Its proof, however, varies slightly from that of Lemma 8.1 in the sensethat R z should be replaced by R u + z and Φ by U . In addition, we recall Remark11.3 ( ii ) to the effect that the application of R u + z increases weak differentiabilityby two units.For the proof of Lemma 8.5, we extensively used that Q commutes with R z .However, on notes that Q does not commute with R u + z . In fact, one has[ R u + z , Q ] = R u + z Q − Q R u + z = R u + z ( Qu ) R u + z , recalling our convention to denote the operator of multipliying with the function x ( Qu )( x ) by ( Qu ). Due to this lack of commutativity, the proof of the analogto Lemma 8.5 is more involved and expanding the resolvent R u + z in the way donein (12.2), the terms discussed in Lemma 8.5 turn out to be the leading terms in apower series expression: Lemma 12.11.
Assume Hypothesis , let z ∈ Σ δ ,ϑ , and suppose that U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is τ -admissible ( cf. Definition , with u as in (12.1) , and C =( Q U ) . Let L = Q + U be given by (7.1) and χ Λ as in (7.3) , Λ > . For z ∈ Σ δ ,ϑ , Λ > , define ξ Λ ( z ) := χ Λ tr b n d (cid:0) [ Q , U ( CR u + z ) n ] (cid:1) and e ξ Λ ( z ) := χ Λ tr b n d (cid:0) [ Q , Q ( CR u + z ) n ] (cid:1) . Then for all z ∈ Σ δ ,ϑ , the operators ξ Λ ( z ) , e ξ Λ ( z ) are trace class and the families { z tr L ( R n ) ( ξ Λ ( z )) } Λ > and (cid:8) z tr L ( R n ) (cid:0)e ξ Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) .Proof. As in the proof of Lemma 8.5, we start out with ξ Λ ( z ) and observe with(11.1), ξ Λ ( z ) = χ Λ tr b n d (cid:0) [ Q , U ( CR u + z ) n ] (cid:1) = χ Λ tr b n d (cid:18)(cid:20) Q , U (cid:18) C ∞ X k =0 ( R z ( u − k R z (cid:19) n (cid:21)(cid:19) = χ Λ tr b n d (cid:18)(cid:20) Q , U ∞ X k =0 X k ,...,k n kk + ... + k n = k (cid:18) C ( R z ( u − k R z C × ( R z ( u − k R z · · · C ( R z ( u − k n R z (cid:19)(cid:21)(cid:19) HE CALLIAS INDEX FORMULA REVISITED 99 = ∞ X k =0 X k ,...,k n kk + ... + k n = k χ Λ tr b n d (cid:18)(cid:20) Q , U × (cid:18) C ( R z ( u − k R z · · · C ( R z ( u − k n R z (cid:19)(cid:21)(cid:19) . In the expression for ξ Λ ( z ) just derived, we note that the summand for k = 0 hasbeen discussed in Lemma 8.5, so we are left with showing the trace class propertyfor the summands belonging to k >
0. Moreover, we need to derive an estimateguaranteeing that the sum in the expression for ξ Λ ( z ) converges in B . Let k ∈ N > and k , . . . , k n ∈ N > such that k + . . . + k n = k , and consider S k ,...,k n := χ Λ tr b n d (cid:18)(cid:20) Q , U (cid:18) C ( R z ( u − k R z · · · C ( R z ( u − k n R z (cid:19)(cid:21)(cid:19) = χ Λ tr b n d (cid:18) Q U (cid:18) C ( R z ( u − k R z · · · C ( R z ( u − k n R z (cid:19) − U (cid:18) C ( R z ( u − k R z · · · C ( R z ( u − k n R z (cid:19) Q (cid:19) . (12.5)Let j ∈ { , . . . , n } be the smallest index for which k j >
1. Then the first summandin (12.5) reads T := Q U ( CR z ) j − C ( R z ( u − k j R z · · · C ( R z ( u − k n R z = Q U ( CR z ) j − CR z (( u − R z ) k j · · · ( CR z )(( u − R z ) k n = Q U ( CR z ) j (( u − R z ) k j · · · ( CR z )(( u − R z ) k n . (12.6)From Q U ( CR z ) j = U Q ( CR z ) j + [ Q , U ] ( CR z ) j = U (cid:18) j X ℓ =1 ( CR z ) ℓ − [ Q , C ] R z ( CR z ) j − ℓ + ( CR z ) j Q (cid:19) + [ Q , U ] ( CR z ) j , one infers Q U ( CR z ) j ∈ B ( n +1) /j , by Lemma 4.5 and the H¨older-type inequality for the Schatten class operators,Theorem 4.2. On the right-hand side of (12.6), apart from ( CR z ) j , there are n − j factors of the form CR z ∈ B n +1 . In addition, there is at least one factor( u − R z ∈ B n +1 , by Lemma 4.5 and the fact that ( u − ∈ L n +1 ( R n ) (as ( u − k T k B kQ U ( CR z ) j k B n +1) /j k CR z k n − j B n +1 k ( u − R z k B n +1 − k . The second term under the trace sign in the expression for S k ,...,k n (see (12.5)) canbe dealt with similarly, so there exists κ > > z ∈ Σ δ ,ϑ ,and k ∈ N , such that k S k ,...,k n k B κ − k .
00 F. GESZTESY AND M. WAURICK
Hence, for all Λ > z ∈ Σ δ ,ϑ one gets k ξ Λ ( z ) k B k ψ Λ ( z ) k B + ∞ X k =1 X k ,...,k n kk + ... + k n = k k S k ,...,k n k B k ψ Λ ( z ) k B + ∞ X k =1 κ ( k + 1) n − k , (12.7)where ψ Λ ( z ) is defined in Lemma 8.5. Inequality (12.7) yields the assertion for ξ Λ .A similar reasoning – as in Lemma 8.5 for e ψ Λ ( z ) – applies to e ξ Λ ( z ). (cid:3) Next, we turn to the proof of a modified version of Lemma 8.6:
Lemma 12.12.
Assume Hypothesis . Let z ∈ Σ δ ,ϑ , and assume that U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is τ -admissible ( cf. Definition , with u as in (12.1) , C = ( Q U ) .Let L = Q + U be given by (7.1) and χ Λ as in (7.3) , Λ > . For z ∈ Σ δ ,ϑ , Λ > ,define ζ Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , U (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR u + z ) n +1 (cid:3)(cid:1) and e ζ Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , (cid:0) Q (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) ( CR u + z ) n +1 (cid:1)(cid:3)(cid:1) . Then for all z ∈ Σ δ ,ϑ ∩ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) , the operators ζ Λ ( z ) , e ζ Λ ( z ) are traceclass and there exists δ ∈ ( δ , such that the families { Σ δ,ϑ ∋ z z tr L ( R n ) ( ζ Λ ( z )) } Λ > and (cid:8) Σ δ,ϑ ∋ z z tr L ( R n ) (cid:0)e ζ Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) .Proof. On can follow the proof of Lemma 8.6 line by line upon replacing Φ by U and R z by R u + z . (We recall CR u + z ∈ B n +1 with k CR u + z k B n +1 k CR z k B n +1 ). (cid:3) As in the derivation of Theorem 7.1 we summarize the results obtained for localboundedness in a theorem (cf. Theorem 8.7):
Theorem 12.13.
Assume Hypothesis , z ∈ Σ δ ,ϑ . Let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be τ -admissible ( cf. Definition , with u as in (12.1) , C = ( Q U ) . Let L = Q + U begiven by (7.1) and χ Λ as in (7.3) , Λ > . Define for z ∈ Σ δ ,ϑ ∩ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , ι Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:3)(cid:1) , and e ι Λ ( z ) := χ Λ tr b n d (cid:0)(cid:2) Q , Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:3)(cid:1) . Then for all z ∈ Σ δ ,ϑ ∩ ̺ ( − LL ∗ ) ∩ ̺ ( − L ∗ L ) , the operators ι Λ ( z ) , e ι Λ ( z ) are traceclass and there exists δ ∈ ( δ , such that the families { Σ δ,ϑ ∋ z z tr L ( R n ) ( ι Λ ( z )) } Λ > and (cid:8) Σ δ,ϑ ∋ z z tr L ( R n ) (cid:0)e ι Λ ( z ) (cid:1)(cid:9) Λ > are locally bounded ( cf. (8.1)) .Proof. As in the proof for Theorem 12.13, it suffices to realize that ι Λ ( z ) = 2 ξ Λ ( z )+ ζ Λ ( z ) and e ι Λ ( z ) = 2 e ξ Λ ( z ) + e ζ Λ ( z ) with the functions introduced in Lemmas 12.11and 12.12. Thus, the assertion follows from the Lemmas 12.11 and 12.12. (cid:3) HE CALLIAS INDEX FORMULA REVISITED 101
The proof of the result analogous to Lemma 8.10 needs some modifications. Inparticular, one should pay particular attention to the assertion concerning h ,j : InLemma 8.10 we proved that h ,j vanishes on the diagonal. Here, we are only ableto give an estimate. Lemma 12.14.
Assume Hypothesis . Let z ∈ Σ δ ,ϑ , and assume that U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) is τ -admissible ( cf. Definition , with u as in (12.1) , C = ( Q U ) .Let L = Q + U be given by (7.1) , R u + z as in (11.1) as well as Q , and γ j,n , j ∈ { , . . . , n } , given by (6.3) , and as in Remark , respectively. Then for n > ,the integral kernel h ,j ( z ) of b n d (cid:0) γ j,n U ( R u + z C ) n − R u + z (cid:1) satisfies, h ,j ( z )( x, x ) = h ,j ( z )( x, x ) + g ,j ( z )( x, x ) , where h ,j ( z ) is the integral kernel of b n d (cid:0) γ j,n U C n − R nu + z (cid:1) and g ,j ( z ) satisfies sup z ∈ Σ δ ,ϑ | g ,j ( z )( x, x ) | κ (1 + | x | ) − n − ε . for all x ∈ R n and some κ > .In addition, if n > and z ∈ R , then the integral kernel h ,j ( z ) of tr b n d (cid:0) γ j,n Q ( R u + z C ) n − R u + z (cid:1) satisfies sup z ∈ Σ δ ,ϑ | h ,j ( z )( x, x ) | κ (1 + | x | ) − n . Proof.
We start with h ,j ( z ). Using the Neumann series expression in (11.1), onecomputes, H ,j ( z ) := tr b n d (cid:0) γ j,n Q ( R u + z C ) n − R u + z (cid:1) = tr b n d (cid:18) γ j,n Q (cid:18) ∞ X k =0 ( R z ( u − k R z C (cid:19) n − ∞ X k =0 ( R z ( u − k R z (cid:19) = tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + tr b n d (cid:18) γ j,n Q (cid:18) ∞ X k =1 ( R z ( u − k R z C (cid:19) × (cid:18) ∞ X k =0 ( R z ( u − k R z C (cid:19) n − × ∞ X k =0 ( R z ( u − k R z (cid:19) + · · · + tr b n d (cid:18) γ j,n Q (cid:18) ∞ X k =0 ( R z ( u − k R z C (cid:19) n − × ∞ X k =1 ( R z ( u − k R z (cid:19) = tr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) + tr b n d (cid:0) γ j,n Q ( R z ( u − R u + z C ) n − R u + z (cid:1) + · · · + tr b n d (cid:0) γ j,n Q ( R u + z C ) n − ( R z ( u − R u + z (cid:1) .
02 F. GESZTESY AND M. WAURICK
By Lemma 8.10, the diagonal of the integral kernel associated withtr b n d (cid:0) γ j,n Q ( R z C ) n − R z (cid:1) vanishes. Thus, it remains to address the asymptotics of the diagonal of the integralkernel associated withtr b n d (cid:0) γ j,n Q ( R z ( u − R u + z C ) n − R u + z (cid:1) + · · · + tr b n d (cid:0) γ j,n Q ( R u + z C ) n − ( R z ( u − R u + z (cid:1) . One observes that the function ( u −
1) vanishes outside B (0 , τ ) ⊂ R n (we recallHypothesis 12.6). Being bounded by 1, it particularly satisfies the estimate | ( u − x ) | (1 + τ ) n (1 + | x | ) − n , x ∈ R n . Realizing that the function C is bounded, the assertion for h ,j ( z ) follows fromRemark 11.10.The assertion about h ,j can be shown with Remark 5.18 (replacing the operators R µ in that remark by R u + z and using that the integral kernel of R u + z can beestimated by the respective one for R z ) , see Theorem 11.8) and the asymptoticconditions imposed on U (see Definition 12.5). (cid:3) The analog of Theorem 8.11, stated below, is now shown in the same way, em-ploying Theorems 11.11 and 11.8:
Theorem 12.15.
Assume Hypothesis , z ∈ Σ δ ,ϑ . Let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be τ -admissible ( cf. Definition , with u as in (12.1) , C = ( Q U ) . Let L = Q + U be given by (7.1) , R u + z as in (11.1) as well as Q , and γ j,n , j ∈ { , . . . , n } , givenby (6.3) , and as in Remark , respectively. Then there exists z > , such thatfor all z ∈ C with Re( z ) > z , the integral kernels g and g of the operators tr b n d (cid:0) γ j,n U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) and tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) , respectively, satisfy for some κ > , ( | g ( x, x ) | + | g ( x, x ) | ) κ (1 + | x | ) − n , x ∈ R n . The next result, the analog of Corollary 8.12, is slightly different compared tothe previous analogs since the operator tr b n d (cid:0) γ j,n U C n − R n z (cid:1) is not replaced bytr b n d (cid:0) γ j,n U C n − R nu + z (cid:1) . Indeed, Corollary 8.12 was used to show that the only im-portant term for the computation for the index is given by tr b n d (cid:0) γ j,n U C n − R n z (cid:1) ,for which we computed the integral over the diagonal of the corresponding integralkernel in Proposition 8.13, eventually yielding the formula for the index. Since theasserted formulas for admissible and τ -admissible potentials are the same, we needto have a result to the effect that the integral of over the diagonal of the integralkernels of the operators tr b n d (cid:0) γ j,n U C n − R nu + z (cid:1) and tr b n d (cid:0) γ j,n U C n − R n z (cid:1) shouldlead to the same results. In fact, this is part of the proof of the following result: Corollary 12.16.
Assume Hypothesis , z ∈ Σ δ ,ϑ . Let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be τ -admissible ( cf. Definition , with u as in (12.1) , C = ( Q U ) . Let L = Q + U be given by (7.1) , R u + z as in (11.1) as well as Q , and γ j,n , j ∈ { , . . . , n } , givenby (6.3) , and as in Remark , respectively. ( i ) Let n ∈ N > , j ∈ { , . . . , n } . Then there exists z > , such that if z ∈ C , HE CALLIAS INDEX FORMULA REVISITED 103
Re( z ) > z , and h and g denote the integral kernel of b n d (cid:0) γ j,n U C n − R n z (cid:1) and J jL ( z ) , respectively, then for some κ > , | h ( x, x ) − g ( x, x ) | κ (1 + | x | ) − n − ε , x ∈ R n . ( ii ) The assertion of part ( i ) also holds for n = 3 , if, in the above statement, J jL ( z ) is replaced by J jL ( z ) − d ( γ j, Q R z CR z ) .Proof. One recalls from Lemma 12.8, J jL ( z ) = 2 tr b n d (cid:0) γ j,n Q ( R u + z C ) n − R u + z (cid:1) + 2 tr b n d (cid:0) γ j,n U ( R u + z C ) n − R u + z (cid:1) + tr b n d (cid:0) γ j,n Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) + tr b n d (cid:0) γ j,n U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) n (cid:1) . With the help of Theorem 12.15 one deduces that the integral kernels of the lasttwo terms may be estimated by κ (1 + | x | ) − n on the diagonal. The integral kernelof the first term is also bounded by κ ′ (1 + | x | ) − n for a suitable κ ′ by Lemma 12.14.Hence, it remains to inspect the second term on the right-hand side. The assertionfollows from Lemma 12.14 once we establish estimates for the respective integralkernels of the differencestr b n d (cid:0) γ j,n U C n − R nu + z (cid:1) − tr b n d (cid:0) γ j,n U C n − R n z (cid:1) (12.8)and tr d ( γ j, Q R u + z CR u + z ) − tr d ( γ j, Q R z CR z ) . (12.9)In this context we will use the equation R u + z = ∞ X k =0 R z (( u − R z ) k = R z + ∞ X k =1 R z (( u − R z ) k = R z + R z ( u − R u + z . Thus, (12.9) and (12.8) readtr d ( γ j, Q R z ( u − R u + z CR z ) + tr d ( γ j, Q R z CR z ( u − R u + z ) ++ tr d ( γ j, Q R z ( u − R u + z CR z ( u − R u + z ) (12.10)and n X k =1 tr b n d (cid:0) γ j,n U C n − R k − u + z R z ( u − R u + z R n − ku + z (cid:1) , (12.11)respectively. In each summand of (12.10) and (12.11) there is one term ( u −
1) whichis compactly supported and thus clearly satisfies for some κ ′ > | ( u − x ) | κ ′ (1 + | x | ) − n , x ∈ R n . Hence, the assertion on the asymptotics of the integralkernels associated with the operators in (12.9) and (12.8) follows from Corollary11.9. (cid:3) We are now ready to prove Theorem 10.2 for
R >
Theorem 12.17 (Theorem 10.2 for
R >
0, smooth case) . Let n, d ∈ N , n > odd,and Φ ∈ C ∞ b (cid:0) R n , C d × d (cid:1) satisfy the following assumptions: ( i ) Φ( x ) = Φ( x ) ∗ , x ∈ R n .
04 F. GESZTESY AND M. WAURICK ( ii ) There exist c > and R > such that | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ) . ( iii ) There exists ε > / such that for all α ∈ N n there is κ > with k ∂ α Φ( x ) k κ ( (1 + | x | ) − , | α | = 1 , (1 + | x | ) − − ε , | α | > , x ∈ R n . Let e L = Q + Φ as in (7.1) , δ ∈ ( − , , ϑ ∈ (0 , π/ . Then there exists τ > suchthat for all τ -admissible potentials U with U = sgn(Φ) on sufficiently large balls,and with L := Q + U , the following assertions ( α ) – ( δ ) hold: ( α ) There exists δ δ < and < ϑ ϑ such that for all Λ > the family Σ δ,ϑ ∋ z zχ Λ tr b n d (( L ∗ L + z ) − − ( LL ∗ + z ) − ) ∈ B ( L ( R n )) (12.12) is analytic. ( β ) The family { f Λ } Λ > of holomorphic functions f Λ : Σ δ,ϑ ∋ z tr (cid:0) zχ Λ tr b n d (( L ∗ L + z ) − − ( LL ∗ + z ) − ) (cid:1) (12.13) is locally bounded ( see (8.1)) . ( γ ) The limit f := lim Λ →∞ f Λ exists in the compact open topology and satisfies forall z ∈ Σ δ,ϑ , f ( z ) = c n (1 + z ) − n/ lim Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr( U ( x )( ∂ i U ( x ) . . . ( ∂ i n − U )( x )) x j d n − σ ( x ) , (12.14) where c n := 12 (cid:18) i π (cid:19) ( n − / n − / . ( δ ) the operators e L and L are Fredholm operators and ind (cid:0)e L (cid:1) = ind( L ) = f (0)= c n lim Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − (12.15) × ˆ Λ S n − tr( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x j d n − σ ( x ) . As mentioned earlier in connection with the proof of Theorem 12.17, we willfollow the analogous reasoning used for the proof for Theorem 7.1. So for the proofof Theorem 12.17 we now need to replace the statements Theorem 7.8, Lemma 7.7,Lemma 8.10, Theorem 8.7 and Corollary 8.12 by the respective results Theorem12.10, Lemma 12.8, Lemma 12.14, Theorem 12.13 and Corollary 12.16 obtained inthis section. Since large parts of the proof would just be a repetition of argumentsused in the proof of Theorem 7.1, we will not give a detailed proof for the case n >
5. However, in Section 8, we only sketched how the result for n = 3 comesabout. As this case is notationally less messy, we will now give the full proof for thecase n = 3. As in Section 9, the core idea is to regularize the expressions involved HE CALLIAS INDEX FORMULA REVISITED 105 by multiplying B L ( z ) with (1 − µ ∆) − , µ >
0, from either side, which results in B L,µ ( z ) = (1 − µ ∆) − B L ( z ) (1 − µ ∆) − , (compare with (9.1)), and similarly for J jL,µ ( z ) and A L,µ ( z ), recalling (9.2) and(9.3), respectively. Proof of Theorem , n = 3 . Part ( α ): This follows from Theorem 12.10.Part ( β ): Again by Theorem 12.10, the expression tr( χ Λ B L ( z )), with B L ( z ) asgiven in (3.2), can be computed as the integral over the diagonal of its integralkernel. Next, we denote by A and J the integral kernels for the operators A L ( z )and 2 B L ( z ) − A L ( z ), respectively, and correspondingly A µ and J µ for A L,µ ( z ) and2 B L,µ ( z ) − A L,µ ( z ), µ >
0. Hence, Proposition 5.5 applied to A yields,2 f Λ ( z ) = 2 tr( χ Λ B L ( z ))= ˆ B (0 , Λ) A ( x, x ) + J ( x, x ) d x = ˆ B (0 , Λ) J ( x, x ) d x = lim µ → ˆ B (0 , Λ) J µ ( x, x ) d x, where in the last equality we used the continuity the integral kernels of 2 B L ( z ) and A L ( z ), as well as Lemma 8.9.Next, appealing to the analog result of Lemma 9.2 for Φ being replaced by the τ -admissible potential U , one arrives at2 f Λ ( z ) = lim µ → ˆ B (0 , Λ) J µ ( x, x ) d x = lim µ → ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 [ ∂ j , J jL,µ ( z )] δ { x } (cid:29) d x. Denote K L,µ := (cid:8) x g jL,µ ( z )( x, x ) (cid:9) j ∈{ , , } , where g jL,µ ( z ) is the integral kernel of J jL,µ ( z ), j ∈ { , , } , and K L,z that for (cid:8) J jL ( z ) − d ( γ j, Q R z CR z ) (cid:9) j ∈{ , , } . Invoking Lemmas 9.4 and 8.9, and hence the fact that { x K L,µ ( x, x ) } µ> islocally bounded, one obtainslim µ → ˆ B (0 , Λ) J L,µ ( x, x ) d x = lim µ → ˆ Λ S (cid:16) K L,µ ( x, x ) , x Λ (cid:17) d σ ( x )= ˆ Λ S lim µ → (cid:16) K L,µ ( x, x ) , x Λ (cid:17) d σ ( x )= ˆ Λ S (cid:16) K L,z ( x, x ) , x Λ (cid:17) d σ ( x ) . Hence, one arrives at2 f Λ ( z ) = ˆ Λ S (cid:16) K L,z ( x ) , x Λ (cid:17) d σ ( x ) (12.16)= ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 [ ∂ j , e J jL ( z )] δ { x } (cid:29) d x,
06 F. GESZTESY AND M. WAURICK with e J jL ( z ) := J jL ( z ) − d ( γ j, Q R z CR z ) . For proving that { f Λ } Λ > is locally bounded, we recall from Lemma 12.8 that e J jL ( z ) = 2 tr d (cid:0) γ j, Q ( R u + z C ) R u + z (cid:1) − d ( γ j, Q R z CR z )+ 2 tr d (cid:0) γ j, U ( R u + z C ) R u + z (cid:1) + tr d (cid:0) γ j, Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) (cid:1) + tr d (cid:0) γ j, U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) (cid:1) , and, thus, X j =1 (cid:2) ∂ j , e J jL ( z ) (cid:3) = X j =1 (cid:2) ∂ j , (cid:0) d (cid:0) γ j, Q ( R u + z C ) R u + z (cid:1) − d ( γ j, Q R z CR z ) (cid:1)(cid:3) + X j =1 (cid:2) ∂ j , d (cid:0) γ j, U ( R u + z C ) R u + z (cid:1)(cid:3) + tr d (cid:0)(cid:2) Q , U (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) (cid:3)(cid:1) + tr d (cid:0)(cid:2) Q , (cid:0) Q (cid:0) ( L ∗ L + z ) − + ( LL ∗ + z ) − (cid:1) ( CR u + z ) (cid:1)(cid:3)(cid:1) . Hence,2 f Λ ( z ) = ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 (cid:20) ∂ j , (cid:0) d (cid:0) γ j, Q ( R u + z C ) R u + z (cid:1) − d ( γ j, Q R z CR z ) (cid:1)(cid:21) δ { x } (cid:29) d x + ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 (cid:2) ∂ j , d (cid:0) γ j, U ( R u + z C ) R u + z (cid:1)(cid:3) δ { x } (cid:29) d x + tr( ι Λ ( z )) + tr( e ι Λ ( z )) , where ι Λ ( z ) and e ι Λ ( z ) are defined in Theorem 12.13. With the help of part ( α ),Theorem 12.13 and Lemma 8.3, to prove part ( β ), it suffices to prove the localboundedness of z ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 (cid:2) ∂ j , (cid:0) d (cid:0) γ j, Q ( R u + z C ) R u + z (cid:1) − d ( γ j, Q R z CR z ) (cid:1)(cid:3) δ { x } (cid:29) d x and z ˆ B (0 , Λ) (cid:28) δ { x } , X j =1 (cid:2) ∂ j , d (cid:0) γ j, U ( R u + z C ) R u + z (cid:1)(cid:3) δ { x } (cid:29) d x, both considered as families of functions indexed by Λ >
0. Appealing to Gauss’divergence theorem, it suffices to show that the integral kernels associated with the
HE CALLIAS INDEX FORMULA REVISITED 107 operators Q R u + z CR u + z − Q R z CR z = Q R z ( u − R u + z CR u + z + Q R u + z CR z ( u − R u + z + Q R z ( u − R u + z CR z ( u − R u + z and U ( R u + z C ) R u + z can be estimated on the diagonal by κ ′ (1 + | x | ) − for some κ ′ > | x | . However, this is a consequence of Corollary 11.9, proving part ( β ).Part ( γ ): By Montel’s Theorem, there exists a sequence { Λ k } k ∈ N of positive realstending to infinity such that f := lim k →∞ f Λ k exists in the compact open topology.From (12.16), one recalls2 f Λ ( z ) = ˆ Λ S (cid:16) K L,z ( x, x ) , x Λ (cid:17) d σ ( x ) , with K L,z denoting the integral kernel of (cid:8) J jL ( z ) − d ( γ j, Q R z CR z ) (cid:9) j ∈{ , , } . Next, we choose z > ii ) and let z ∈ Σ z ,ϑ . With h j ,the integral kernel of 2tr d (cid:0) γ j, Ψ C R z (cid:1) , we define H z := ( x h j ( x, x )) j ∈{ , , } .Due to Corollary 12.16 ( ii ) one can find κ > k ∈ N , (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ k S (cid:18) ( K J,z − H z )( x ) , x Λ k (cid:19) R n d σ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Λ k S k ( K J,z − H z )( x ) k R n d σ ( x ) κ ˆ Λ k S (1 + | x | ) − − ε d σ ( x )= κ Λ k ω (1 + Λ k ) − − ε . Hence, lim k →∞ ˆ Λ k S (cid:18) ( K J,z − H z )( x ) , x Λ k (cid:19) R n d σ ( x ) = 0 , and2 f ( z ) = lim k →∞ ˆ Λ k S (cid:18) K J,z ( x ) , x Λ k (cid:19) R n d σ ( x )= lim k →∞ ˆ Λ k S (cid:18) H z ( x ) , x Λ k (cid:19) R n d σ ( x )= (cid:18) i π (cid:19) (1 + z ) − / lim k →∞ ˆ Λ k S × X j =1 (cid:18) X i ,i =1 ε ji i tr( U ( x )( ∂ i U )( x )( ∂ i U )( x ) (cid:1)(cid:19)(cid:18) x j Λ k (cid:19) d σ ( x ) , (12.17)where, for the last integral, we used Proposition 8.13. Theorem 3.4 implies f (0) =ind( L ). In particular, any sequence { Λ k } k ∈ N of positive reals converging to infinity
08 F. GESZTESY AND M. WAURICK contains a subsequence { Λ k ℓ } ℓ such that for that particular subsequence the limitlim ℓ →∞ ˆ Λ kℓ S X j,i ,i =1 ε ji i tr( U ( x )( ∂ i U )( x )( ∂ i U )( x ) (cid:1)(cid:18) x j Λ k ℓ (cid:19) d σ ( x )exists and equals 2 ind( L )[ i/ (8 π )] (1 + z ) − / . (12.18)Hence, the limitlim Λ →∞ ˆ Λ S X j,i ,i =1 ε ji ...i tr( U ( x )( ∂ i U )( x )( ∂ i U )( x )) (cid:18) x j Λ (cid:19) d σ ( x ) (12.19)exists and equals the number in (12.18). On the other hand, for Re( z ) > z with z > i )) the family { f Λ } Λ > converges for Λ → ∞ on the domain C Re >z ∩ Σ δ,ϑ if and only if the limit in (12.19)exists. Indeed, this follows from the explicit expression for the limit in (12.17).Therefore, { f Λ } Λ > converges in the compact open topology on C Re >z ∩ Σ δ,ϑ .By the local boundedness of the latter family on Σ δ,ϑ , the principle of analyticcontinuation for analytic functions implies that the latter family actually convergeson the domain Σ δ,ϑ in the compact open topology. In particular,2 f ( z )(1 + z ) / i/ (8 π )= lim Λ →∞ ˆ Λ S X j,i ,i =1 ε ji ...i tr( U ( x )( ∂ i U )( x )( ∂ i U )( x )) (cid:18) x j Λ (cid:19) d σ ( x ) . Part ( δ ): The Fredholm property of L and e L follows from Lemma 13.3, and theequality ind (cid:0)e L (cid:1) = ind( L ) follows from Theorem 12.2 and Remark 12.4. The re-maining equality in (12.15) follows from part ( γ ) and Theorem 3.4. (cid:3) HE CALLIAS INDEX FORMULA REVISITED 109
The proof of Theorem : The General Case
The strategy to prove Theorem 10.2 for potentials which are only C consistsin an additional convolution with a suitable mollifier, applying Theorem 10.2 (i.e.,Theorem 12.17) for the C ∞ -case, and to use suitable perurbation theorems for theFredholm index. The next result gathers information on mollified functions. Proposition 13.1.
Let n, d ∈ N , Φ ∈ C b (cid:0) R n ; C d × d (cid:1) . Assume ζ ∈ C ∞ ( R n ) with ζ > , supp( ζ ) ⊂ B (0 , , ´ R n ζ ( x ) d n x = 1 and for γ > define ζ γ := γ − n ζ (1 /γ · ) and Φ γ := ζ γ ∗ Φ . ( i ) For all < γ < and j ∈ { , . . . , n } one has k Φ( x ) − Φ γ ( x ) k γ sup y ∈ B (0 ,γ ) k Φ ′ ( x + y ) k , x ∈ R n . ( ii ) Assume, in addition, Φ ∈ C b (cid:0) R n ; C d × d (cid:1) and for some ε > / and all α ∈ N n , | α | = 2 , that for some κ > , k ∂ α Φ( x ) k κ (1 + | x | ) − − ε , x ∈ R n . Then for all α ∈ N n with | α | > , < γ < , k ( ∂ α Φ γ )( x ) k k ∂ α − β ζ k ∞ v n γ −| α | κ (1 − γ + | x | ) − − ε , x ∈ R n , with v n the n -dimensional volume of the unit ball in R n and β ∈ N n such that ( α − β ) ∈ N n and | β | = 2 . ( iii ) If Φ( x ) = Φ( x ) ∗ for all x ∈ R n , then Φ γ ( x ) = Φ γ ( x ) ∗ for all x ∈ R n , γ > . ( iv ) If there exist c > and R > such that | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ) ,then there exists γ > such that for all < γ < γ | Φ γ ( x ) | > ( c/ I d , x ∈ R n \ B (0 , R ) . Proof. ( i ) In order to prove the first inequality, let x ∈ R n , 0 < γ <
1. Then onecomputes k Φ( x ) − Φ γ ( x ) k = (cid:13)(cid:13)(cid:13)(cid:13) ˆ R n (Φ( x ) − Φ( x − y )) ζ γ ( y ) d n y (cid:13)(cid:13)(cid:13)(cid:13) ˆ B (0 , k (Φ( x ) − Φ( x − γy )) k ζ ( y ) d n y (cid:18) ˆ B (0 , ζ ( y ) d n y (cid:19) γ sup y ∈ B (0 ,γ ) k Φ ′ ( x + y ) k . ( ii ) Let 0 < γ < α ∈ N n with | α | >
2, and let β ∈ N n with | β | = 2 be suchthat ( α − β ) ∈ N n . Then for x ∈ R n , k ( ∂ α Φ γ )( x ) k = k ( ∂ α ( ζ γ ∗ Φ)) ( x ) k = (cid:13)(cid:13)(cid:13)(cid:13) (cid:0)(cid:0) ∂ α − β ζ γ (cid:1) ∗ (cid:0) ∂ β Φ (cid:1)(cid:1) ( x ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) ˆ R n (cid:0) ∂ α − β ζ γ (cid:1) ( y ) (cid:0) ∂ β Φ (cid:1) ( x − y ) d n y (cid:13)(cid:13)(cid:13)(cid:13) ˆ R n γ − n −| α | +2 (cid:13)(cid:13)(cid:0) ∂ α − β ζ (cid:1) ( y/γ )( ∂ β Φ)( x − y ) (cid:13)(cid:13) d n y ˆ B (0 , γ −| α | (cid:12)(cid:12)(cid:0) ∂ α − β ζ (cid:1) ( y ) (cid:12)(cid:12) (cid:13)(cid:13) ( ∂ β Φ)( x − γy ) (cid:13)(cid:13) d n y
10 F. GESZTESY AND M. WAURICK k ∂ α − β ζ k ∞ ˆ B (0 , γ −| α | κ (1 − γ + | x | ) − − ε d n y k ∂ α − β ζ k ∞ v n γ −| α | κ (1 − γ + | x | ) − − ε . ( iii ) This is clear.( iv ) By part ( i ), there exists γ > x ∈ R n k Φ( x ) − Φ γ ( x ) k < c/ < γ < γ . Let x ∈ R n such that | x | > R . From | Φ( x ) | > cI d it follows that k Φ( x ) − k /c . Hence, (cid:13)(cid:13) Φ( x ) − (Φ( x ) − Φ γ ( x )) (cid:13)(cid:13) / ∞ X k =0 (cid:0) Φ( x ) − (Φ( x ) − Φ γ ( x )) (cid:1) k Φ( x ) − = (cid:0) − Φ( x ) − (Φ( x ) − Φ γ ( x )) (cid:1) − Φ( x ) − = (Φ( x ) − (Φ( x ) − Φ γ ( x ))) − = Φ γ ( x ) − . Using (cid:13)(cid:13) Φ γ ( x ) − (cid:13)(cid:13) c − P ∞ k =0 − k = 2 /c , one deduces with the help of the spectraltheorem that | Φ γ ( x ) | > ( c/ I d . (cid:3) Remark . Let Φ be as in Theorem 10.2. More precisely, let Φ ∈ C b (cid:0) R n ; C d × d (cid:1) be pointwise self-adjoint, suppose that for some c > R > | Φ( x ) | > cI d forall x ∈ R n \ B (0 , R ), and assume there exists ε > /
2, such that for all α ∈ N n thereexists κ > k ∂ α Φ( x ) k κ ( (1 + | x | ) − , | α | = 1 , (1 + | x | ) − − ε , | α | = 2 . By Proposition 13.1, there exists γ > γ ∈ (0 , γ ), Φ γ (definedas in Proposition 13.1) satisfies the assumptions imposed on Φ in Theorem 12.17.Moreover, by Proposition 13.1 ( i ), for some e κ >
0, the estimate k ∂ j Φ( x ) − ∂ j Φ γ ( x ) k e κ (1 + | x | ) − − ε holds for all j ∈ { , . . . , n } , x ∈ R n , 0 < γ <
1. The latter observation will be usedin the proof of the general case of Theorem 10.2. ⋄ For the sake of completeness, we shall also state the Fredholm property for C -potentials: Lemma 13.3.
Let n, d ∈ N , L = Q + Φ as in (7.1) with Φ ∈ C b (cid:0) R n ; C d × d (cid:1) with Φ( x ) = Φ( x ) ∗ , x ∈ R n . Assume that there exist c > and R > suchthat | Φ( x ) | > cI d for all x ∈ R n \ B (0 , R ) , and that ( ∂ j Φ)( x ) → as | x | → ∞ , j ∈ { , . . . , n } . Then L is a Fredholm operator, and there is γ > such that for all γ ∈ (0 , γ ) , ind( L ) = ind( L γ ) , where L γ = Q + Φ γ with Φ γ given as in Proposition .Proof. By Proposition 13.1, L is a Q -compact perturbation of L γ . Moreover, thelatter is a Fredholm operator for all γ ∈ (0 , γ ) for some γ > (cid:0) guaranteeing that for some e c > e R > | Φ γ ( x ) | > e cI d for all x ∈ R n \ B (cid:0) , e R (cid:1)(cid:1) and Lemma 12.1. Thus, the assertion follows from the invariance of the Fredholmindex under relatively compact perurbations. (cid:3) We are prepared to conclude the proof of the main theorem also for C -potentials. Proof of Theorem , the nonsmooth case.
Let ζ ∈ C ∞ ( R n ) be as in Proposition13.1, and define Φ γ as in the latter proposition. Let γ ∈ (0 ,
1) be as in Proposition
HE CALLIAS INDEX FORMULA REVISITED 111 iv ). As observed in Remark 13.2, Φ γ satisfies the assumptions imposed on Φin Theorem 12.17. Next, by Lemma 13.3, L γ := Q + Φ γ is Fredholm and ind( L ) =ind( L γ ).Hence, by the C ∞ -version of Theorem 10.2, that is, by Threorem 12.17, oneinfers thatind ( L γ ) = (cid:18) i π (cid:19) ( n − / n − / Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr (cid:0) sgn(Φ γ ( x ))( ∂ i sgn(Φ γ ))( x ) . . . ( ∂ i n − sgn(Φ γ ))( x ) (cid:1) x j d n − σ ( x ) . It sufices to shown that the limit γ → ω : R > ∋ x x + c − x is decreasing and, denoting k Φ k ∞ := sup x ∈ R n k Φ( x ) k , one gets 0 <ω ( k Φ k ∞ ) ω ( k Φ( x ) k ), x ∈ R n . Let 0 < γ < ω ( k Φ k ∞ ) / (4 κ ) ∧ (1 / ∧ γ . For0 < γ < γ and all x ∈ R n one deduces with the help of Proposition 13.1 ( i ) that k Φ( x ) − Φ γ ( x ) k γ κ (1 − γ + | x | ) − ω ( k Φ k ∞ ) / . Hence, by Theorem 10.3, one gets for some
K > x ∈ R n with | x | > R , k sgn(Φ( x )) − sgn(Φ γ ( x )) k sup T ∈ S | x | > R B (Φ( x ) ,ω (( k Φ k ∞ ) / k sgn ′ ( T ) kk Φ( x ) − Φ γ ( x ) k γK (1 + | x | ) − . Similarly, Proposition 13.1 ( i ) implies for some K ′ > j ∈{ ,...,n } k sgn ′ (Φ( x ))( ∂ j Φ)( x ) − sgn ′ (Φ γ ( x ))( ∂ j Φ)( x ) k max j ∈{ ,...,n } k ( ∂ j Φ)( x ) k sup T ∈ S | x | > R B (Φ( x ) ,ω ( k Φ k ∞ ) / k sgn ′′ ( T ) kk Φ( x ) − Φ γ ( x ) k γK ′ (1 + | x | ) − . For i , . . . , i n − ∈ { , . . . , n } and x ∈ R n , and with the convention ∂ i := 1, onegets for some constants K ′′ , K ′′′ > k (sgn(Φ( x ))( ∂ i sgn(Φ))( x ) . . . ( ∂ i n − sgn(Φ))( x )) − (sgn(Φ γ ( x ))( ∂ i sgn(Φ γ ))( x ) . . . ( ∂ i n − sgn(Φ γ ))( x )) k n − X j =0 (cid:13)(cid:13)(cid:13)(cid:13) j − Y k =0 ( ∂ i k sgn(Φ))( x )(( ∂ i j sgn(Φ))( x ) − ( ∂ i j sgn(Φ γ ))( x )) × n − Y k = j +1 ( ∂ i k sgn(Φ γ ))( x ) (cid:13)(cid:13)(cid:13)(cid:13) K ′′ (1 + | x | ) − n (cid:18) n − X j =1 k ( ∂ i j sgn(Φ))( x ) − ( ∂ i j sgn(Φ γ ))( x ) k + (1 + | x | ) − k sgn(Φ( x )) − sgn(Φ γ ( x )) k (cid:19)
12 F. GESZTESY AND M. WAURICK K ′′ (1 + | x | ) − n (cid:18) n − X j =1 k sgn ′ (Φ( x ))( ∂ i j Φ)( x ) − sgn ′ (Φ γ ( x ))( ∂ i j Φ γ )( x ) k + γK (1 + | x | ) − (cid:19) K ′′ (1 + | x | ) − n (cid:18) n − X j =1 k sgn ′ (Φ( x ))( ∂ i j Φ)( x ) − sgn ′ (Φ γ ( x ))( ∂ i j Φ)( x ) k + n − X j =1 k sgn ′ (Φ γ ( x ))( ∂ i j Φ)( x ) − sgn ′ (Φ γ )( x )( ∂ i j Φ γ )( x ) k + γK (1 + | x | ) − (cid:19) K ′′ (1 + | x | ) − n (cid:18) n − X j =1 γK ′ (1 + | x | ) − + n − X j =1 sup T ∈ S | x | > R B (Φ( x ) ,ω ( k Φ k ∞ ) / k sgn ′ ( T ) kk ( ∂ i j Φ)( x ) − ( ∂ i j Φ γ )( x ) k + γK (1 + | x | ) − (cid:19) K ′′′ γ (1 + | x | ) − n − ε . (13.1)Next, for Λ > φ Λ := 1Λ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr (cid:0) sgn(Φ( x ))( ∂ i sgn(Φ( x ))) . . . ( ∂ i n − sgn(Φ( x ))) (cid:1) x j d n − σ ( x )and φ γ Λ := 1Λ n X j,i ,...,i n − =1 ε ji ...i n − ˆ Λ S n − tr (cid:0) sgn(Φ γ ( x )) × ( ∂ i sgn(Φ γ ( x ))) . . . ( ∂ i n − sgn(Φ γ ( x ))) (cid:1) x j d n − σ ( x ) . It remains to prove that { φ Λ } Λ converges and that its limit coincides with ind( L ).But, with the help of estimate (13.1) one getslim sup Λ →∞ | φ γ Λ − φ Λ | lim sup Λ →∞ ˆ Λ S n − K ′′′ γ (1 + | x | ) − n − ε | x | Λ d n − σ ( x ) = 0 , which implies the remaining assertion. (cid:3) Remark . ( i ) Of course a simple manner in which to invoke less regular po-tentials is the perturbation with compactly supported potentials. Thus, the aboveresult should be read as Φ is C “in a neighborhood of infinity”.( ii ) The formula for the Fredholm index suggests that Theorem 10.2 might be weak-ened in the sense that potentials that are only C should lead to the same result.Our method of proof needs that for some K > γ > (cid:13)(cid:13) ( ∂ i j Φ)( x ) − ( ∂ i j Φ γ )( x ) (cid:13)(cid:13) K (1 + | x | ) − − ε . To prove the latter estimate we needinformation on the second derivative of Φ. ⋄ HE CALLIAS INDEX FORMULA REVISITED 113
We conclude with a nontrivial example of the Fredholm index. In view of thediscussion in Example 4.8 and the erroneous statement in (1.22) this could be thetype of potentials Callias had in mind.
Example 13.5.
Let n = 3 , γ , , γ , , γ , ∈ C × be the corresponding matrices ofthe Euclidean Dirac Algebra ( see Appendix A) . Consider L = Q +Φ as in (7.1) with Φ( x ) := P j =1 γ j, x j | x | − , j ∈ { , , } . Then Φ( x ) = I and Theorem ap-plies. Given formula (10.1) for the Fredholm index, a straightforward computationyields tr (cid:0) Φ( x )( ∂ i Φ)( x )( ∂ i Φ)( x ) (cid:1) = tr (cid:0) γ j, γ i , γ i , x j | x | − (cid:1) , x ∈ R n \{ } , for all j, i , i ∈ { , , } pairwise distinct, and hence, ind( L ) = i π lim Λ →∞ X j,i ,i =1 ε ji i ˆ Λ S tr (cid:0) Φ( x )( ∂ i Φ)( x )( ∂ i Φ)( x ) (cid:1) x j d σ ( x )= i π lim Λ →∞ X j =1 ˆ Λ S i x j Λ x j d σ ( x )= i π lim Λ →∞ ˆ Λ S i d σ ( x )= − π ˆ S d σ ( x ) = − .
14 F. GESZTESY AND M. WAURICK
A Particular Class of Non-Fredholm Operators L and TheirGeneralized Witten Index This section is devoted to a particular non-Fredhom situation and motivated inpart by extensions of index theory for a certain class of non-Fredholm operatorsinitiated in [12], [24], [53] (see also [13], [27]). Here we make the first steps inthe direction of non-Fredholm operators closely related to the operator L in (6.2)studied by Callias [22] and introduce a generalized Witten index.We very briefly outline the idea presented in [53]: Let L be a densely defined,closed, linear operator in a Hilbert space H . Assume that (cid:2) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:3) ∈ B ( H )for one (and hence for all) z ∈ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ), and that the limitind W ( L ) := lim x ↓ + x tr H (cid:0) ( L ∗ L + x ) − − ( LL ∗ + x ) − (cid:1) (14.1)exists. Then ind W ( L ) is called the Witten index of L . In fact, for the special case ofoperators in space dimension n = 1 (with appropriate potential), this limit is easilyshown to exist and to assume values in (1 / Z , see [23]. These examples, however,heavily rely on the fact that the underlying spatial dimension for the operator L equals one.While the Fredholm index is well-known to be invariant with respect to relativelycompact additive perturbations, we emphasize that this cannot hold for the Wittenindex (cf. [12], [53]). In fact, it can be shown that the Witten index is invari-ant under additive perturbations that are relatively trace class (among additionalconditions, see [53] for details).We now provide a further generalization of the Witten index adapted to thenon-Fredholm operators discussed in this section for odd dimensions n >
3. Theabstract set-up reads as follows:
Definition 14.1.
Let L be a densely defined, closed linear operator in H m for some m ∈ N . Assume there exist sequences { T Λ } Λ ∈ N , { S ∗ Λ } Λ ∈ N in B ( H ) converging to I H in the strong operator topology, and denote S Λ := S ∗∗ Λ , Λ ∈ N . In addition, supposethat the map Ω ∋ z T Λ tr m (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) S Λ assumes values in B ( H ) for some open set Ω ⊆ ̺ ( − L ∗ L ) ∩ ̺ ( − LL ∗ ) with (0 , δ ) ⊆ Ω ∩ R for some δ > . Moreover, assume that { f Λ } Λ ∈ N , where f Λ : Ω ∋ z z tr H (cid:0) T Λ tr m (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) S Λ (cid:1) , Λ ∈ N , converges in the compact open topology as Λ → ∞ to some function f : Ω → C andthat f (0 + ) exists. Then we call ind gW,T,S ( L ) := f (0 + ) . (14.2) the generalized Witten index of L ( with respect to T and S ) . If L satisfies the as-sumptions needed for defining ind gW,T,S ( L ) , then we say that L admits a generalizedWitten index . Whenever the sequences { T Λ } Λ ∈ N and { S Λ } Λ ∈ N in Definition 14.1 are clear fromthe context, we will just write ind gW ( · ) instead of ind gW,T,S ( · ). HE CALLIAS INDEX FORMULA REVISITED 115
Remark . We briefly elaborate on some properties of the regularized index justdefined.( i ) It is easy to see that the generalized Witten index is independent of the chosen Ω.Indeed, the main observation needed is that if Ω and Ω satisfy the requirementsimposed on Ω in Definition 14.1, then so does Ω ∩ Ω .( ii ) The generalized Witten index is invariant under unitary equivalence of H .Indeed, let L be a densely defined, closed linear operator in H m for which thegeneralized Witten index exists with respect to { T Λ } Λ ∈ N and { S Λ } Λ ∈ N . Let H beanother Hilbert space and U : H → H an isometric isomorphism. Then e L := U ∗ · · · U ∗ ...... . . .0 · · · U ∗ L U · · · U ...... . . .0 · · · U admits a generalized Witten index, andind gW,T,S ( L ) = ind gW,U ∗ T U,U ∗ SU (cid:0)e L (cid:1) , where, in obvious notation, we used ind gW,U ∗ T U,U ∗ SU (cid:0)e L (cid:1) to denote the generalizedWitten index of e L with respect to { U ∗ T Λ U } Λ ∈ N and { U ∗ S Λ U } Λ ∈ N .For the proof of e L admitting a generalized Witten index, it suffices to observethat for Λ ∈ N and z ∈ Ω,tr H (cid:0) T Λ tr m (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) S Λ (cid:1) = tr H (cid:0) T Λ U U ∗ tr m (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) U U ∗ S Λ (cid:1) = tr H (cid:0) T Λ U tr m (cid:0) ( e L ∗ e L + z ) − − ( e L e L ∗ + z ) − (cid:1) U ∗ S Λ (cid:1) = tr H (cid:0) U ∗ T Λ U tr m (cid:0) ( e L ∗ e L + z ) − − ( e L e L ∗ + z ) − (cid:1) U ∗ S Λ U (cid:1) . ⋄ Remark . The definition of the Witten index in (14.1) suggests introducingthe spectral shift function ξ ( · ; LL ∗ , L ∗ L ) for the pair of self-adjoint operators( LL ∗ , L ∗ L ) and hence to express the Witten index asind W ( L ) = ξ (0 + ; LL ∗ , L ∗ L ) , (14.3)employing the fact (see, e.g., [105, Ch. 8]),tr H (cid:0) f ( L ∗ L ) − f ( LL ∗ ) (cid:1) = ˆ [0 , ∞ ) f ′ ( λ ) ξ ( λ ; LL ∗ , L ∗ L ) dλ, (14.4)for a large class of functions f . The approach (14.3) in terms of spectral shiftfunctions was introduced in [12], [53] (see also [13], [19], [79, Chs. IX, X], [80]) andindependently in [27]. This circle of ideas continues to generate much interest, see,for instance, [23], [24], [52], and the extensive list of references therein. It remainsto be seen if this can be applied to the generalized Witten index (14.2). ⋄ Next, we will construct non-Fredholm Callias-type operators L , which meet theassumptions in the definition for the generalized Witten index, that is, operators L which admit a generalized Witten index. In fact, the theory developed in theprevious chapters provides a variety of such examples (cf. the end of this section inTheorem 14.11).
16 F. GESZTESY AND M. WAURICK
We start with an elementary observation:
Proposition 14.4.
Let n = 2 b n + 1 ∈ N odd. Then Q as in (4.1) with dom( Q ) = H ( R n ) b n as an operator in L ( R n ) b n is non-Fredholm.Proof. It suffices to observe that the symbol of Q is a continuous function vanishingat 0. (cid:3) The fundamental result leading to Theorem 14.11 is contained in the followinglemma.
Lemma 14.5.
Let n = 2 b n + 1 ∈ N odd, d ∈ N , Q as in (6.3) , Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) pointwise self-adjoint, and let L = Q + Φ be as in (7.1) . Assume that there exists P = P ∗ = P ∈ C d × d \{ } such that for all x ∈ R n , P Φ( x ) = Φ( x ) P = 0 . Define P := I L ( R n ) b n ⊗ P and denote H P := L ( R n ) b n ⊗ ran( P ) . Then L and L ∗ leave the space H P invariant. Moreover, L is unitarily equivalent to (cid:18) ι ∗ P Q ι P ι ∗ P ⊥ Q ι P ⊥ + ι ∗ P ⊥ Φ ι P ⊥ (cid:19) with ι P and ι P ⊥ the canonical embeddings from H P and H ⊥P into L ( R n ) b n d , re-spectively.Remark . Recalling Q given as in (4.1), we claim that in the situation of Lemma14.5, ι ∗ P Q ι P = Q ⊗ I ran P . Indeed, equality is plain when applied to C ∞ -functions and thus the general casefollows by a closure argument. For the closedness of ι ∗ P Q ι P we use Lemma 14.5: L is closed and, by unitary equivalence, so is (cid:18) ι ∗ P Q ι P ι ∗ P ⊥ Q ι P ⊥ + ι ∗ P ⊥ Φ ι P ⊥ (cid:19) . (14.5)Hence, the diagonal entries of the closed block operator matrix in (14.5), and thus ι ∗ P Q ι P , are closed. ⋄ In order to prove Lemma 14.5, we invoke some auxiliary results of a generalnature. The first two (Lemmas 14.7 and 14.8) are concerned with commutativityproperties of the operator Q . Lemma 14.7.
Let n, m ∈ N , P ∈ C m × m , j ∈ { , . . . , n } . Then ( I L ( R n ) ⊗ P ) ∂ j ⊆ ∂ j ( I L ( R n ) ⊗ P ) , where ∂ j : H j ( R n ) m ⊆ L ( R n ) m → L ( R n ) m is the distributional derivative withrespect to the j th variable and, H j ( R n ) is the space of L -functions whose derivativewith respect to the j th variable can be represented as an L -function.Proof. Clearly,( I L ( R n ) ⊗ P ) ∂ j φ = ∂ j ( I L ( R n ) ⊗ P ) φ, φ ∈ C ∞ ( R n ) m . Next, the operator ∂ j ( I L ( R n ) ⊗ P ) is closed, hence,( I L ( R n ) ⊗ P ) ∂ j ⊆ ( I L ( R n ) ⊗ P ) ∂ j ⊆ ∂ j ( I L ( R n ) ⊗ P ) , yields the assertion. (cid:3) HE CALLIAS INDEX FORMULA REVISITED 117
Lemma 14.8.
Let n, d ∈ N , n = 2 b n or n = 2 b n + 1 for some b n ∈ N . Let Q as in (6.3) ( defined in L ( R n ) b n d ) . Let P ∈ C d × d and denote P := I L ( R n ) b n ⊗ P . Then, PQ ⊆ QP . Proof.
We note that for all j ∈ { , . . . , n } and γ j,n as in Section A, γ j,n P = P γ j,n ,where we viewed γ j,n ∈ B ( L ( R n ) b n ). Hence, using dom( Q ) = T nj =1 dom( ∂ j ),Lemma 14.7 implies PQ = P n X j =1 γ j,n ∂ j = n X j =1 P γ j,n ∂ j = n X j =1 γ j,n P ∂ j ⊆ n X j =1 γ j,n ∂ j P = QP . (cid:3) Before turning to the proof of Lemma 14.5, we recall a general result on therepresentability of operators as block operator matrices (the same calculus has alsobeen employed in [84, Lemma 3.2]):
Lemma 14.9.
Let P ∈ B ( H ) be an orthogonal projection, W : D ( W ) ⊆ H → H closed and linear. Assume that P W ⊆ W P and ( I H − P ) W ⊆ W ( I H − P ) . Denote by ι P : ran( P ) → H and ι P ⊥ : ker( P ) → H the canonical embeddings, re-spectively. Then W is unitarily equivalent to a block operator matrix. More pre-cisely, (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) W (cid:0) ι P ι P ⊥ (cid:1) = (cid:18) ι ∗ P W ι P ι ∗ P ⊥ W ι P ⊥ (cid:19) (14.6) with ι ∗ P W ι P and ι ∗ P ⊥ W ι P ⊥ closed linear operators.Proof. First, one observes that the operators (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) and (cid:0) ι P ι P ⊥ (cid:1) are unitaryand inverses of each other. Moreover, it is plain that a block diagonal operatormatrix is closed if and only if its diagonal entries are closed. Thus, as W is closedby hypothesis, it suffices to prove equality (14.6). One notes that P = ι P ι ∗ P andsimilarly, ( I H − P ) = ι P ⊥ ι ∗ P ⊥ , and hence computes, W = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) W (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P Wι ∗ P ⊥ W (cid:19) (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι P ι ∗ P Wι ∗ P ⊥ ι P ⊥ ι ∗ P ⊥ W (cid:19) (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) ⊆ (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W ι P ι ∗ P ι ∗ P ⊥ W ι P ⊥ ι ∗ P ⊥ (cid:19) (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W ι P ι ∗ P ι P ι ∗ P ⊥ W ι P ⊥ ι ∗ P ⊥ ι P ⊥ (cid:19) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W ι P ι ∗ P ⊥ W ι P ⊥ (cid:19) (cid:18) ι ∗ P ι ∗ P ⊥ (cid:19)
18 F. GESZTESY AND M. WAURICK = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W ι P ι ∗ P ι ∗ P ⊥ W ι P ⊥ ι ∗ P ⊥ (cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W ι P ι ∗ P (cid:0) ι P ι ∗ P + ι P ⊥ ι ∗ P ⊥ (cid:1) ι ∗ P ⊥ W ι P ⊥ ι ∗ P ⊥ (cid:0) ι P ι ∗ P + ι P ⊥ ι ∗ P ⊥ (cid:1)(cid:19) = (cid:0) ι P ι P ⊥ (cid:1) (cid:18) ι ∗ P W (cid:0) ι P ι ∗ P + ι P ⊥ ι ∗ P ⊥ (cid:1) ι ∗ P ⊥ W (cid:0) ι P ι ∗ P + ι P ⊥ ι ∗ P ⊥ (cid:1)(cid:19) ⊆ W, concluding the proof. (cid:3) At this instant we are in a position to prove Lemma 14.5.
Proof of Lemma . By Lemma 14.8, P L ⊆ L P and ( I L ( R n ) b nd − P ) L ⊆ L ( I L ( R n ) b nd − P ) . Hence, by Lemma 14.9, L is unitarily equivalent to (cid:18) ι ∗ P Lι P ι ∗ P ⊥ Lι P ⊥ . (cid:19) The assertion, thus, follows from ι ∗ P Φ ι P = 0 (valid by hypothesis). (cid:3) From Proposition 14.4 and Lemma 14.5 one infers the following result.
Corollary 14.10.
Let n = 2 b n + 1 ∈ N odd, d ∈ N , assume Q is given by (6.3) , Φ ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) pointwise self-adjoint, and let L = Q +Φ be as in (7.1) . Assumethat there exists P = P ∗ = P ∈ C d × d \{ } such that P Φ( x ) = Φ( x ) P = 0 , x ∈ R n . Then L is non-Fredholm.Proof. Define H P := L ( R n ; C b n ⊗ ran( P )), denote the embedding from H P into L ( R n ) b n d by ι P , and denote the embedding from H ⊥ P into L ( R n ) b n d by ι P ⊥ . ByLemma 14.5, the operator L is unitarily equivalent to (cid:18) ι ∗ P Q ι P ι ∗ P ⊥ Q ι P ⊥ + ι ∗ P ⊥ Φ ι P ⊥ (cid:19) , which by Remark 14.6 may also be written as (cid:18) Q ⊗ I ran P Q ⊗ I ker P + ι ∗ P ⊥ Φ ι P ⊥ (cid:19) . In particular, σ ( L ) = σ (cid:0) Q ⊗ I ran P (cid:1) ∪ σ (cid:0) Q ⊗ I ker P + ι ∗ P ⊥ Φ ι P ⊥ (cid:1) . Since ran( P ) is at least one-dimensional, it follows from Proposition 14.4 that 0 ∈ σ ess (cid:0) Q ⊗ I ran P (cid:1) . Hence, 0 ∈ σ ess ( L ), implying that L is non-Fredholm. (cid:3) We conclude this section with non-trivial examples illustrating the generalizedWitten index introduced in this section:
HE CALLIAS INDEX FORMULA REVISITED 119
Theorem 14.11.
Assume Hypothesis and let U ∈ C ∞ b (cid:0) R n ; C d × d (cid:1) be a τ -admissible potential. Let ℓ ∈ N and define Φ : R n → C ( d + ℓ ) × ( d + ℓ ) , x (cid:18) U ( x ) (cid:19) . Let L := Q + Φ , as in (6.2) . Then the following assertions ( α ) – ( δ ) hold: ( α ) For all Λ > , the family Σ ,ϑ ∋ z zχ Λ tr b n ( d + ℓ ) (( L ∗ L + z ) − − ( LL ∗ + z ) − ) ∈ B ( L ( R n )) (14.7) is analytic. ( β ) The family { f Λ } Λ > of analytic functions f Λ : Σ ,ϑ ∋ z tr (cid:0) zχ Λ tr b n ( d + ℓ ) (( L ∗ L + z ) − − ( LL ∗ + z ) − ) (cid:1) (14.8) is locally bounded ( see (8.1)) . ( γ ) The limit f := lim Λ →∞ f Λ exists in the compact open topology and satisfies forall z ∈ Σ ,ϑ , f ( z ) = c n (1 + z ) − n/ lim Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − × ˆ Λ S n − tr( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x j d n − σ ( x ) , (14.9) where c n := 12 (cid:18) i π (cid:19) ( n − / n − / . ( δ ) L is non-Fredholm, it admits a generalized Witten index, given by ind gW ( L ) = f (0 + )= c n lim Λ →∞ n X j,i ,...,i n − =1 ε ji ...i n − (14.10) × ˆ Λ S n − tr( U ( x )( ∂ i U )( x ) . . . ( ∂ i n − U )( x )) x j d n − σ ( x ) , which is actually an integer as it coincides with the Fredholm index of e L := Q + U in L ( R n ) b n d , that is, ind gW ( L ) = ind (cid:0)e L (cid:1) . (14.11) Proof.
The proof rests on Theorem 12.17, Lemma 14.5, Remark 14.6, and specifi-cally, for the assertion that L is non-Fredholm, on Corollary 14.10. Indeed, invokingLemma 14.5 and Remark 14.6 with P = (cid:18) I ℓ
00 0 (cid:19) ∈ C ( d + ℓ ) × ( d + ℓ ) , one computes, recalling e L = Q + U , L ∗ L = ( −Q + Φ)( Q + Φ)= (cid:18) −Q −Q + U (cid:19) (cid:18) Q Q + U (cid:19)
20 F. GESZTESY AND M. WAURICK = (cid:18) − ∆ 00 e L ∗ e L (cid:19) . A similar computation applies to LL ∗ . One deduces for z ∈ C Re > ,(( L ∗ L + z ) − − ( LL ∗ + z ) − )= (cid:18) ( − ∆ + z ) −
00 ( e L ∗ e L + z ) − (cid:19) − (cid:18) ( − ∆ + z ) −
00 ( e L e L ∗ + z ) − (cid:19) = (cid:18) e L ∗ e L + z ) − − ( e L e L ∗ + z ) − (cid:19) . Thus,tr b n ( d + ℓ ) (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − ) (cid:1) = tr b n d (cid:0) ( e L ∗ e L + z ) − − ( e L e L ∗ + z ) − (cid:1) . Hence, the assertions (14.7)–(14.10) indeed follow from Theorem 12.17 applied to e L . (cid:3) HE CALLIAS INDEX FORMULA REVISITED 121
Appendix A. Construction of the Euclidean Dirac Algebra
For a concise presentation of the construction of the Euclidean Dirac algebra asa specific case of Clifford algebras, see, for instance, [93, Chapter 11].
Definition A.1.
Given two matrices A = ( a ij ) i,j ∈{ ,...,n } ∈ C n × n and B =( b ij ) i,j ∈{ ,...,m } ∈ C m × m , one defines their Kronecker product A ◦ B by A ◦ B := a B a B · · · a n Ba B . . . ...... a n B · · · a nn B = (cid:0) a ⌈ pm ⌉⌈ qm ⌉ b ((( p −
1) mod m )+1)((( q −
1) mod m )+1) (cid:1) p,q ∈{ ,...,mn } ∈ C nm × nm , where ⌈ x ⌉ := min { z ∈ Z | z > x } for all x ∈ R and k mod ℓ denotes the nonnegativeinteger j ∈ { , . . . , ℓ − } such that k − j is divisible by ℓ , with ℓ, k ∈ Z . Proposition A.2.
Let n, m, ℓ, k ∈ N , A ∈ C n × n , B ∈ C m × m , C ∈ C ℓ × ℓ , D ∈ C k × k . Then one concludes that A ◦ ( B ◦ C ) = ( A ◦ B ) ◦ C, ( A ◦ B ) ∗ = A ∗ ◦ B ∗ , tr( A ◦ B ) = tr( A ) tr( B ) , if n = m and ℓ = k then, AB ◦ CD = ( A ◦ C ) ( B ◦ D ) . Proof.
We only sketch a proof for the first assertion. It boils down to the followingequations, & (cid:6) jk (cid:7) m ' = (cid:24) jm k (cid:25) , (cid:18)(cid:18)(cid:24) jk (cid:25) − (cid:19) mod m (cid:19) + 1 = (cid:24) ( j − mk ) + 1 k (cid:25) , ( j − mk ) mod k = j − k, j ∈ { , . . . , mnk } . The expressions on the left-hand side correspond to the indices of the entries of
A, B and C , respectively, in ( A ◦ B ) ◦ C and, similarly, the expressions on the right-hand sides correspond to the respective indices of the entries of A , B and C in A ◦ ( B ◦ C ). (cid:3) Definition A.3.
Introduce the Pauli matrices σ := (cid:18) (cid:19) , σ := (cid:18) − ii (cid:19) , σ := (cid:18) − (cid:19) , in addition, define γ , := σ , γ , := σ . Let b n ∈ N . Recursively, one sets γ k, b n +1 := γ k, b n , k ∈ { , . . . , b n } ,γ b n +1 , b n +1 := ( − i ) b n γ , b n · · · γ b n, b n ,
22 F. GESZTESY AND M. WAURICK and γ k, b n +2 := σ ◦ γ k, b n , k ∈ { , . . . , b n } ,γ b n +1 , b n +2 := i b n σ ◦ ( γ , b n · · · γ b n, b n ) ,γ b n +2 , b n +2 := σ ◦ I b n , with I r the identity matrix in C r , r ∈ N .Remark A.4 . By induction, one obtains γ k, b n , γ k, b n +1 , γ b n +1 , b n +1 ∈ C b n × b n , k ∈ { , . . . , b n } . (A.1) ⋄ Lemma A.5.
Let γ , . . . , γ k ∈ B ( K ) for some Hilbert space K and such that for all j, k ∈ { , . . . , k } , j = k, one has γ j γ k + γ k γ j = 0 . Then γ k γ k − · · · γ = ( − k ( k − / γ γ · · · γ k . Proof.
The assertion being obvious for k = 1, we assume that the assertion of thelemma holds for some k ∈ N . Then γ k +1 γ k γ k − · · · γ = ( − k γ k γ k − · · · γ γ k +1 = ( − [ k ( k − / k γ γ · · · γ k γ k +1 = ( − k ( k +1) / γ γ · · · γ k γ k +1 . (cid:3) Corollary A.6.
For all k, l ∈ { , . . . , n } , n ∈ N > , one has γ k,n γ l,n + γ l,n γ k,n = 2 δ kl I b n , where γ j,n is given in Definition A.3 , j ∈ { , . . . , n } , and b n ∈ N is such that n = 2 b n or n = 2 b n + 1 .Proof. The assertion holds for n = 2 . Assume that the assertion is valid for n = 2 b n for some b n ∈ N . Then Lemma A.5 implies γ b n +1 , b n +1 = ( − i ) b n ( γ , b n · · · γ b n, b n ) ( γ , b n · · · γ b n, b n )= ( − b n ( γ , b n · · · γ b n, b n ) ( − b n (2 b n − / ( γ b n, b n · · · γ , b n )= ( − b n +2 b n − b n I b n = I b n . For k ∈ { , . . . , b n − } one computes γ k, b n +1 γ b n +1 , b n +1 = γ k, b n ( − i ) b n ( γ , b n · · · γ b n, b n )= ( − b n − ( − i ) b n ( γ , b n · · · γ b n, b n ) γ k, b n = − γ b n +1 , b n +1 γ k, b n +1 . Hence, the assertion is established for γ k, b n +1 , k ∈ { , . . . , b n + 1 } .For k, l ∈ { , . . . , b n } one computes with the help of Proposition A.2, γ k, b n +2 γ l, b n +2 + γ l, b n +2 γ k, b n +2 = ( σ ◦ γ k, b n ) ( σ ◦ γ l, b n )+ ( σ ◦ γ l, b n ) ( σ ◦ γ k, b n )= σ ◦ γ k, b n γ l, b n + σ ◦ γ l, b n γ k, b n = I ◦ ( γ k, b n γ l, b n + γ l, b n γ k, b n )= I ◦ δ kl I b n = 2 δ kl I b n +1 . HE CALLIAS INDEX FORMULA REVISITED 123
One observes that γ b n +1 , b n +2 = (cid:16) i b n (cid:17) σ ◦ ( γ , b n · · · γ b n, b n ) = (cid:16) i b n (cid:17) σ ◦ ( − i ) − b n I b n = (cid:16) i b n (cid:17) σ ◦ ( − − b n (cid:16) i b n (cid:17) − I b n = I b n +1 , using γ b n +1 , b n +1 = I b n . Moreover, γ b n +2 , b n +2 = σ ◦ I b n = I b n +1 . In addition, onenotes that γ b n +2 , b n +2 γ b n +1 , b n +2 = σ i b n σ ◦ γ , b n · · · γ b n, b n = − i b n σ σ ◦ γ , b n · · · γ b n, b n = − γ b n +1 , b n +2 γ b n +2 , b n +2 ,γ b n +2 , b n +2 γ k, b n +2 = σ σ ◦ γ k, b n = − γ k, b n +2 γ b n +2 , b n +2 , and γ b n +1 , b n +2 γ k, b n +2 = i b n σ σ ◦ γ , b n · · · γ b n, b n γ k, b n = σ i b n σ ◦ ( − b n − γ k, b n γ , b n · · · γ b n, b n = − γ k, b n +2 γ b n +1 , b n +2 for all k ∈ { , . . . , b n } , implying the assertion. (cid:3) Corollary A.7.
For all k ∈ N , n ∈ N > , and k n , one has γ ∗ k,n = γ k,n , where γ j,n is given in Definition A.3 , j ∈ { , . . . , n } .Proof. We will proceed by induction. Before doing so, we note that due to CorollaryA.6 and Lemma A.5, for all k ∈ { , . . . , n } , γ k,n γ k − ,n · · · γ ,n = ( − k ( k − / γ ,n γ ,n · · · γ k,n . One observes that γ , and γ , are self-adjoint. We assume that γ k, b n is self-adjointfor all k ∈ { , . . . , b n } for some b n ∈ N . The only matrices not obviously self-adjointusing the induction hypothesis and Proposition A.2 are γ b n +1 , b n +2 and γ b n +1 , b n +1 .Since the proof for either case follows along similar lines, it suffices to prove theself-adjointness of γ b n +1 , b n +2 . For this purpose one computes, γ ∗ b n +1 , b n +2 = (cid:16) i b n σ ◦ ( γ , b n · · · γ b n, b n ) (cid:17) ∗ = i b n ( − b n σ ∗ ◦ ( γ , b n · · · γ b n, b n ) ∗ = i b n ( − b n σ ◦ ( γ b n, b n · · · γ , b n )= i b n ( − b n +[2 b n (2 b n − / σ ◦ ( γ , b n · · · γ b n, b n )= i b n ( − b n +2 b n − b n σ ◦ ( γ , b n · · · γ b n, b n )= γ b n +1 , b n +2 . (cid:3) Next, we proceed to establish the following result on traces:
Proposition A.8.
Let b n ∈ N and suppose that γ j, b n , γ j ′ , b n +1 , j ∈ { , . . . , b n } , j ′ ∈ { , . . . , b n + 1 } , are given as in Definition A.3 . Then, tr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 (cid:1) = 0 , if i , . . . , i k +1 ∈ { , . . . , b n + 1 } and k < b n , tr (cid:0) γ i , b n · · · γ i k +1 , b n (cid:1) = 0 , if i , . . . , i k +1 ∈ { , . . . , b n } and k ∈ N , tr (cid:0) γ i , b n +1 · · · γ i b n +1 , b n +1 (cid:1) = (2 i ) b n ε i ··· i b n +1 , if i , . . . , i b n +1 ∈ { , . . . , b n + 1 } ,
24 F. GESZTESY AND M. WAURICK where ε i ··· i b n +1 is the fully anti-symmetric symbol in b n + 1 dimensions, that is, ε i ...i b n +1 = 0 whenever |{ i , . . . , i b n +1 }| < b n + 1 and if π : { , . . . , b n + 1 } →{ , . . . , b n + 1 } is bijective, then ε π (1) ··· π (2 b n +1) = sgn( π ) . Proof.
The first formula can be seen as follows. Since k < b n , there exists i ∈{ , . . . , b n + 1 }\{ i , . . . , i k +1 } , and one computestr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 (cid:1) = tr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 γ i, b n +1 (cid:1) = tr (cid:0) γ i, b n +1 γ i , b n +1 · · · γ i k +1 , b n +1 γ i, b n +1 (cid:1) = − tr (cid:0) γ i , b n +1 γ i, b n +1 · · · γ i k +1 , b n +1 γ i, b n +1 (cid:1) = . . . = ( − k +1 tr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 γ i, b n +1 γ i, b n +1 (cid:1) = − tr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 (cid:1) . Hence, tr (cid:0) γ i , b n +1 · · · γ i k +1 , b n +1 γ i, b n +1 γ i, b n +1 (cid:1) = 0.The second assertion can be proved along the same lines.The third assertion follows upon taking into account the cancellation and anti-commuting properties of the algebra in conjunction with the first statement, oncethe following equality has been established:tr ( γ , b n +1 · · · γ b n +1 , b n +1 ) = (2 i ) b n . To verify the latter identity one computestr ( γ , b n +3 · · · γ b n +3 , b n +3 )= tr (cid:16) γ , b n +2 · · · γ b n +2 , b n +2 ( − i ) b n +1 γ , b n +2 · · · γ b n +2 , b n +2 (cid:17) = ( − i ) b n +1 tr (cid:16) ( σ ◦ γ , b n ) · · · ( σ ◦ γ b n, b n ) i b n ( σ ◦ γ , b n . . . γ b n, b n ) ( σ ◦ I b n ) × ( σ ◦ γ , b n ) · · · ( σ ◦ γ b n, b n ) i b n ( σ ◦ γ , b n . . . γ b n, b n ) ( σ ◦ I b n ) (cid:17) = (cid:0) i (cid:1) b n ( − i ) b n +1 tr (cid:16) σ b n +11 σ σ b n +11 σ ◦ ( γ , b n . . . γ b n, b n ) (cid:17) = ( − b n +1 ( − i ) b n +1 tr ( σ σ σ σ ◦ I b n ) = i b n +1 b n +1 . (cid:3) We conclude with the following result.
Corollary A.9.
Let n ∈ N > be odd, V be a complex vector space, k ∈ N , with k +1 < n , i , . . . , i k ∈ { , . . . , n } . Let Φ : { , . . . , n } n → V be satisfying the property X ( i k +3 ,...,i n ) ∈{ ,...,n } n − k − Φ( i , . . . , i k , i, j, i k +3 , . . . , i n )= X ( i k +3 ,...,i n ) ∈{ ,...,n } n − k − Φ( j , . . . , j k , j, i, i k +3 , . . . , i n ) , i, j ∈ { , . . . , n } . Then X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k tr (cid:0) γ i ,n · · · γ i n ,n (cid:1) Φ( i , . . . , i n ) = 0 , where γ j,n , j ∈ { , . . . , n } , are given by Definition A.3 .Proof.
In the course of this proof we shall suppress the index n in γ i,n . X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k γ i · · · γ i n Φ( i , . . . , i k , i k +1 , i k +2 , i k +3 , . . . , i n ) HE CALLIAS INDEX FORMULA REVISITED 125 = 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k γ i · · · γ i n Φ( i , . . . , i k , i k +1 , i k +2 , i k +3 , . . . , i n )+ 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k γ i · · · γ i n × Φ( i , . . . , i k , i k +2 , i k +1 , i k +3 , . . . , i n )= 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k (cid:0) γ i · · · γ i k γ i k +1 γ i k +2 γ i k +3 · · · γ i n + γ i · · · γ i k γ i k +2 γ i k +1 γ i k +3 · · · γ i n ,n (cid:1) × Φ( i , . . . , i n )= 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k ,i k +1 = i k +2 (cid:0) γ i · · · γ i k γ i k +1 γ i k +2 γ i k +3 · · · γ i n + γ i · · · γ i k γ i k +2 γ i k +1 γ i k +3 · · · γ i n (cid:1) × Φ( i , . . . , i n )+ 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k ,i k +1 = i k +2 (cid:0) γ i · · · γ i k γ i k +1 γ i k +2 γ i k +3 · · · γ i n + γ i · · · γ i k γ i k +2 γ i k +1 γ i k +3 · · · γ i n (cid:1) × Φ( i , . . . , i n )= 12 X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k ,i k +1 = i k +2 (cid:0) γ i · · · γ i k γ i k +3 · · · γ i n + γ i · · · γ i k γ i k +3 · · · γ i n (cid:1) Φ( i , . . . , i n )= X ( i k +1 ,i k +2 ,i k +3 ,...,i n ) ∈{ ,...,n } n − k ,i k +1 = i k +2 γ i · · · γ i k γ i k +3 · · · γ i n Φ( i , . . . , i n ) . Applying the internal trace to the latter sum, one infers that each term vanishesby Proposition A.8. (cid:3)
26 F. GESZTESY AND M. WAURICK
Appendix B. A Counterexample to [22, Lemma 5]In this appendix we shall provide a counterexample for the trace class propertyasserted in [22, Lemma 5]. The counterexample is constructed in dimension n = 3and recorded in Theorem B.5.Analogously to Example 4.8, we let Φ assume values in the 2 × σ j , j ∈ { , , } . Beforewe give an explicit formula for Φ, we need the following definitions. Let φ ∈ C ∞ ( R )be a function interpolating between 0 and 1 with φ ( x ) = ( , x , , x > , x ∈ R , φ := φ ( − ( · + 1))and let φ ,r,t := φ (cid:0) t − ( · ) − t − r (cid:1) , φ ,r,t := φ (cid:0) t − ( · ) − t − r (cid:1) , r, t > . For r , r , t , t ∈ (0 , ∞ ) with r + t < r − t , this yields the following variant ofa smooth “cut-off” function ψ r ,r ,t ,t := φ ,r ,t φ ,r − t ,t . (B.1)One notes that ψ r ,r ,t ,t ∈ C ∞ ( R ). We will use the following properties of ψ r ,r ,t ,t (all of them are easily checked):0 ψ r ,r ,t ,t , (B.2) ψ r ,r ,t ,t | [ r + t ,r − t ] = 1 , (B.3) ψ r ,r ,t ,t | R \ [ r ,r ] = 0 , (B.4) | ψ ′ r ,r ,t ,t | d (cid:18) t ∨ t (cid:19) on [ r , r + t ] ∪ [ r − t , r ] , (B.5) | ψ ( ℓ ) r ,r ,t ,t | d ℓ (cid:18) t ℓ ∨ t ℓ (cid:19) , ℓ ∈ N > , (B.6)with d := k φ ′ k ∞ := sup x ∈ R | φ ′ ( x ) | and d ℓ := k φ ( ℓ )1 k ∞ , ℓ ∈ N > . For k ∈ N > define r k := k − X j =1 j = 2 k − ,ψ ,k := ψ r k ,r k +1 , k , k , ψ ,k := ψ r k ,r k +1 , k , k . One observes that r k + 12 2 k = r k +1 −
12 2 k < r k +1 −
120 2 k , r k + 136 2 k = r k +1 − k < r k +1 − k , so that ψ ,k and ψ ,k are well-defined. For x = ( x , x , x ) ∈ R we let Φ : R → C × be defined as follows,Φ( x ) := X j =1 σ j + ∞ X k =2 k / X j =1 σ j ξ k,j ( x ) , x ∈ R , (B.7)where ξ k,j ( x ) := 1 r k +1 ψ ,k ( | x | )( x j − r k ) ψ ,k ( x j ) , x ∈ R . (B.8) HE CALLIAS INDEX FORMULA REVISITED 127
One observes that Φ ∈ C ∞ ( R ). Next, we introduce the sets B k := (cid:8) x ∈ R (cid:12)(cid:12) r k | x | r k +1 (cid:9) ∩ [ j ∈{ , , } (cid:8) x ∈ R (cid:12)(cid:12) r k x j r k +1 (cid:9) , k ∈ N , (B.9)and e B k := (cid:26) x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) r k + 12 2 k | x | r k +1 −
120 2 k (cid:27)\ (cid:26) x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) r k + 136 2 k x , x , x r k +1 − k (cid:27) , k ∈ N . (B.10)Before turning to the properties of Φ, we study ξ k,j first. Lemma B.1.
Let j ∈ { , , } , ℓ ∈ { , , } , ξ k,j as in (B.8) , B k , e B k as in (B.9) and (B.10) , respectively. Then the following assertions ( α ) – ( γ ) hold: ( α ) For all k ∈ N , x ∈ R , ξ k,j ( x ) = 0 implies x ∈ B k . (B.11)( β ) For all α ∈ N , there exists κ > such that for all k ∈ N , | ∂ α ξ k,j ( x ) | κ (1 + | x | ) −| α | , x ∈ B k . (B.12)( γ ) For all ℓ ∈ { , , } , and all k ∈ N , ∂ ℓ ξ k,j ( x ) = δ ℓj , x ∈ e B k . (B.13) Proof. (B.11): The assertion follows from (B.4) and the definition of B k .(B.12): One observes that ψ ,k = 0 on ( r k , r k +1 ) and that 0 ψ ,k | ( x j − r k ) ψ ,k ( x j ) | k , j ∈ { , , } , k ∈ N > . One recalls, r k = k − X j =1 j = 2 k − < r k +1 = 2 k +1 − k − , in particular, (1 /r k +1 ) κ − k for some κ >
0. Hence, (cid:13)(cid:13)(cid:13)(cid:13) r k +1 ψ ,k ( | x | ) X j =1 σ j ( x j − r k ) ψ ,k ( x j ) (cid:13)(cid:13)(cid:13)(cid:13) χ B k ( x ) κ , x ∈ R , with B k introduced in (B.9). Thus, (B.12) holds for ℓ = 0. Next, for the firstpartial derivatives in item (B.12) one computes for ℓ = j ,( ∂ ℓ ξ k,j )( x ) = 1 r k +1 ψ ′ ,k ( | x | ) x ℓ | x | ( x j − r k ) ψ ,k ( x j )and for ℓ = j ,( ∂ j ξ k,j )( x ) = 1 r k +1 ψ ′ ,k ( | x | ) x j | x | ( x j − r k ) ψ ,k ( x j ) + 1 r k +1 ψ ,k ( | x | ) σ j ψ ,k ( x j )+ 1 r k +1 ψ ,k ( | x | ) σ j ( x j − r k ) ψ ′ ,k ( x j ) , j ∈ { , , } . For x ∈ B k , one has | ( x j − r k ) ψ ′ k ( x j ) | c by (B.5), | ψ ′ k ( | x | )( x ℓ − r k ) ψ k ( x ℓ ) | c by(B.2) and (B.5) and for some κ, c > k ∈ N > , (cid:13)(cid:13)(cid:13)(cid:13) r k +1 ψ k ( | x | ) σ j ψ k ( x j ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) r k +1 ψ k ( | x | ) (cid:13)(cid:13)(cid:13)(cid:13) κ (1 + | x | ) −
128 F. GESZTESY AND M. WAURICK since for all x ∈ B k one has r k +1 > | x | . Higher-order derivatives can be treatedsimilarly, using (B.6), proving assertion (B.12).(B.13): This is obvious. (cid:3) The next lemma gives an account of the asymptotic properties of Φ and itsderivatives.
Lemma B.2.
Let Φ be given by (B.7) . Then the following assertions ( α ) – ( γ ) hold: ( α ) Φ is bounded, pointwise self-adjoint, Φ ∈ C ∞ (cid:0) R ; C × (cid:1) , Φ( x ) − exists for all x ∈ R , and Φ( x ) −→ | x |→∞ I . ( β ) There exists κ > such that | ( ∂ j Φ)( x ) | κ (1 + | x | ) − , x ∈ R , j ∈ { , , } , and the formula ( ∂ j Φ)( x ) = k − / σ j x ∈ e B k , j ∈ { , , } , k ∈ N , holds, where e B k is given by (B.10) . ( γ ) For all α ∈ N n with | α | > , there exists κ ′ > , such that | ( ∂ α Φ)( x ) | κ ′ (1 + | x | ) −| α | , x ∈ R . Proof.
For item ( α ), we use Lemma B.1 (B.12) with ℓ = 0 together with the factthat B k ∩ B k ′ = ∅ for k ′ > k +1, so Φ is bounded. Φ is easily verified to be pointwiseself-adjoint. For showing invertibility of Φ, one computes for x ∈ B k ,Φ( x )Φ( x ) = (cid:18) X j =1 (cid:18) σ j + 1 k / ξ k,j ( x ) σ j (cid:19) (cid:19) = X j =1 (cid:18) k / ξ k,j ( x ) (cid:19) I = X j =1 k / ξ k,j ( x ) + (cid:18) k / ξ k,j ( x ) (cid:19) ! I > I , implying ( α ). Item ( β ) follows from Lemma B.1, (B.12), and (B.13), whereas item( γ ) follows from (B.1), (B.12). (cid:3) In order to prove that tr (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) for L = Q + Φ (with Q as in (6.3)) is not trace class for z in a neighborhood of 0, we need to invoke thefollowing general statement: Theorem B.3 ([16, Theorem 3.1]) . Let K ∈ B ( L ( R n )) be an operator inducedby a continuous integral kernel k : R n × R n → C . Assume that K ∈ B (cid:0) L ( R n ) (cid:1) .Then the function x k ( x, x ) defines an element of L ( R n ) . Before we state and prove the main result of this section, we need to study thevolume of e B k : Lemma B.4.
Let e B k , k ∈ N , be as in (B.10) . Then there exists k ∈ N , such thatfor all k ∈ N > k , vol (cid:0) e B k (cid:1) = 2 k / (36) . HE CALLIAS INDEX FORMULA REVISITED 129
Proof.
Let k ∈ N . One observes that if x ∈ (cid:26) x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) r k + 136 2 k x , x , x r k +1 − k (cid:27) , then √ (cid:18) r k + 136 2 k (cid:19) | x | √ (cid:18) r k +1 − k (cid:19) . Since 16 / √ /
10, for sufficiently large k ∈ N , the estimates √ (cid:18) r k + 136 2 k (cid:19) > (cid:18) (cid:19) k − >
32 2 k − r k + 12 2 k , and √ (cid:18) r k +1 − k (cid:19) (cid:18) − (cid:19) k − k − r k +1 −
120 2 k , hold. Consequently, for sufficiently large k ∈ N , (cid:26) x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) r k + 136 2 k x , x , x r k +1 − k (cid:27) ⊆ e B k . Hence, there exists k ∈ N , such that for all k ∈ N > k ,vol (cid:0) e B k (cid:1) = (cid:18) r k +1 − k − ( r k + 136 2 k ) (cid:19) . (cid:3) Theorem B.5.
Let n = 3 and Q and Φ be given by (6.3) and (B.7) , respectively.Then there exists δ > such that for L = Q + Φ , and for any real z ∈ B (0 , δ ) \{ } , tr (cid:0) ( L ∗ L + z ) − − ( LL ∗ + z ) − (cid:1) / ∈ B (cid:0) L ( R ) (cid:1) . Proof.
In view of Remark 11.3 and Lemma 7.7 it suffices to check whether or not e T := tr (cid:0) ( R z C ) R z (cid:1) is a trace class operator, where C = [ Q , Φ], and R z are given by (2.2) and (4.6),respectively.Arguing by contradiction, we shall assume that e T ∈ B (cid:0) L ( R ) (cid:1) . One observes,( R z C ) R z = R z CR z CR z CR z = [ R z , C ] R z CR z CR z + CR z R z CR z CR z = [ R z , C ] R z CR z CR z + CR z [ R z , C ] R z CR z + CR z CR z R z CR z = [ R z , C ] R z CR z CR z + CR z [ R z , C ] R z CR z + CR z CR z [ R z , C ] R z + CR z CR z CR z R z . (B.14)By Lemmas 4.5 and B.2, one gets CR z , R z C ∈ B (cid:0) L ( R ) (cid:1) and [ R z , C ] ∈B (cid:0) L ( R ) (cid:1) . Hence, by Theorem 4.2, one infers that despite the last term in (B.14),all operators are trace class. In addition, one computes CR z CR z CR z R z = C [ R z , C ] R z CR z R z + C R z R z CR z R z (B.15)= C [ R z , C ] R z CR z R z + C R z [ R z , C ] R z R z + C R z CR z
30 F. GESZTESY AND M. WAURICK = C [ R z , C ] R z CR z R z + C R z [ R z , C ] R z R z + C [ R z , C ] R z + C R z . (B.16)Next, one notes that Lemma 4.4 implies the relation[ R z , C ] = R z (∆ C ) R z + 2 R z ( Q C ) Q R z . With the help of Lemma B.2, there exists κ > {k C ( x ) k , k (∆ C )( x ) k , k ( Q C )( x ) k} κ (1 + | x | ) − , x ∈ R . Therefore, Lemma 4.5 and Theorem 4.2 imply C [ R z , C ] R z CR z = C ( R z (∆ C ) R z + 2 R z ( Q C ) Q R z ) R z CR z = CR z (∆ C ) R z CR z + 2 CR z ( Q C ) R z Q R z CR z ∈ B · B · B + B · B · B · B ⊆ B , and, C R z [ R z , C ] R z ∈ B · B · B ⊆ B , as well as, C [ R z , C ] R z = C R z ((∆ C ) R z + 2 R z ( Q C ) Q R z ) R z = C R z (∆ C ) R z R z + 2 C R z R z ( Q C ) QR z R z ∈ B · B · B + B · B · B ⊆ B . Noting that the inner trace maps trace class operators to trace class operators (cf.Remark 3.2), and combining (B.14) and (B.16) together with our assumption that e T is trace class, one concludes that T := tr (cid:0) C R z (cid:1) = tr (cid:0) C (cid:1) R z ∈ B (cid:0) L ( R ) (cid:1) . Next, one observes that T is an integral operator induced by the following integralkernel t : ( x, y ) ˆ ( R ) tr (cid:0) C (cid:1) ( x ) r z ( x − x ) r z ( x − x ) r z ( x − x ) r z ( x − y ) × d x d x d x , where r z is the Helmholtz Green’s function, see (5.11) associated with ( − ∆ +(1 + z )) − . By Theorem 5.1 (and Proposition 5.4), t is continuous. As T is traceclass, Theorem B.3 implies that the map x t ( x, x ) generates an L ( R )-function.Hence, ˆ R | t ( x, x ) | d x = ˆ R (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( R ) tr (cid:0) C (cid:1) ( x ) r z ( x − x ) r z ( x − x ) r z ( x − x ) r z ( x − x ) × d x d x d x (cid:12)(cid:12)(cid:12)(cid:12) d x = ˆ R (cid:12)(cid:12)(cid:12)(cid:12) ˆ ( R ) tr (cid:0) C (cid:1) ( x ) r z ( x ) r z ( x − x ) r z ( x − x ) r z ( x ) × d x d x d x (cid:12)(cid:12)(cid:12)(cid:12) d x HE CALLIAS INDEX FORMULA REVISITED 131 = ˆ R (cid:12)(cid:12) tr (cid:0) C (cid:1) ( x ) (cid:12)(cid:12) d x (cid:10) δ { } , R z δ { } (cid:11) < ∞ . In other words, tr (cid:0) C (cid:1) ∈ L ( R ) . (B.17)The rest of the proof aims at showing that the statement (B.17) is false. For thispurpose we need to compute tr (cid:0) [ Q , Φ] (cid:1) on S k ∈ N > e B k , with e B k given in (B.10).We recall from Lemma B.2 ( ii ),( ∂ j Φ)( x ) = 1 k / r k +1 σ j , x ∈ e B k , j ∈ { , , } . Hence, tr (cid:0) [ Q , Φ] (cid:1) ( x ) = X j,m,ℓ =1 iε jmℓ tr (cid:0) ( ∂ j Φ)( x )( ∂ m Φ)( x )( ∂ ℓ Φ)( x ) (cid:1) = X j,m,ℓ =1 iε jmℓ k r k +1 tr ( σ j σ m σ ℓ )= − X j,m,ℓ =1 ε jmℓ k r k +1 = −
24 1 k r k +1 , implying, (cid:12)(cid:12) tr (cid:0) [ Q , Φ] (cid:1) ( x ) (cid:12)(cid:12) >
24 1 k r k +1 , x ∈ e B k , k ∈ N > . (B.18)However, employing Lemma B.4 one infers with the help (B.18) that for some k ∈ N , tr (cid:0) C (cid:1) = (cid:13)(cid:13) tr (cid:0) [ Q , Φ] (cid:1)(cid:13)(cid:13) L ( R ) > ∞ X k = k k r k +1 vol (cid:0) e B k (cid:1) = 1(36) ∞ X k = k k r k +1 k = 1(36) ∞ X k = k k k − k = ∞ , contradicting (B.17). (cid:3) Remark
B.6 . It might be of interest to compute the index of Q + Φ, with the poten-tial Φ constructed in this section: One notes that Φ is a Q -compact perturbationof the operator Q + U in L ( R n ) , where U := X j =1 σ j . Since U = I and ∂ j U = 0, j ∈ { , , } , one infers that U is admissible. The indexformula in Theorem 10.1 leads to ind( Q + U ) = 0, and hence to ind( Q + Φ) = 0.
32 F. GESZTESY AND M. WAURICK
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Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
E-mail address : [email protected] URL : Institut f¨ur Analysis, Fachrichtung Mathematik, Technische Universit¨at Dresden,01062 Dresden, Germany
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