The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains
aa r X i v : . [ m a t h - ph ] M a y THE SPECTRAL SHIFT FUNCTION FOR COMPACTLYSUPPORTED PERTURBATIONS OF SCHR ¨ODINGEROPERATORS ON LARGE BOUNDED DOMAINS
PETER D. HISLOP AND PETER M ¨ULLER
Abstract.
We study the asymptotic behavior as L → ∞ of the finite-volume spectral shift function for a positive, compactly-supported perturba-tion of a Schr¨odinger operator in d -dimensional Euclidean space, restrictedto a cube of side length L with Dirichlet boundary conditions. The size ofthe support of the perturbation is fixed and independent of L . We prove thatthe Ces`aro mean of finite-volume spectral shift functions remains pointwisebounded along certain sequences L n → ∞ for Lebesgue-almost every en-ergy. In deriving this result, we give a short proof of the vague convergenceof the finite-volume spectral shift functions to the infinite-volume spectralshift function as L → ∞ . Our findings complement earlier results of W.Kirsch [Proc. Amer. Math. Soc. , 509–512 (1987), Int. Eqns. Op. Th. , 383–391 (1989)] who gave examples of positive, compactly-supportedperturbations of finite-volume Dirichlet Laplacians for which the pointwiselimit of the spectral shift function does not exist for any given positiveenergy. Our methods also provide a new proof of the Birman–Solomyak for-mula for the spectral shift function that may be used to express the measuregiven by the infinite-volume spectral shift function directly in terms of thepotential. Statement of the Problem and Result
The spectral shift function (SSF) plays an important role in scattering theoryfor Schr¨odinger operators [Y]. For the particular case of a quantum mechanicalsystem in a finite volume, the SSF, as a function of energy E , counts the changein the number of eigenvalues below E due to adding a perturbing potential V .We are interested in the following question: given cubes Λ L ⊂ R d in d -dimensional Euclidean space, which are centered at the origin and have sidelengths L >
0, what is the limiting behavior as L → ∞ of the SSF correspondingto the Laplacian plus a background potential H ( L )0 := − (∆ L /
2) + V and itsperturbation H ( L )1 := H ( L )0 + V ? Both operators are defined on the Hilbertspace L (Λ L ) with Dirichlet boundary conditions. The potentials V and V act as multiplication operators corresponding to real-valued functions V and V such that max { , V } ∈ K loc ( R d ) , max { , − V } ∈ K ( R d ) ,V ∈ K loc ( R d ) , V > , supp( V ) is compact . ( ⋆ ) Appeared in:
Proc. Amer. Math. Soc. , 2141–2150 (2010).
P. D. HISLOP AND P. M ¨ULLER
It is understood in ( ⋆ ) that the compact support of V is independent of L , andwe have written K ( R d ) and K loc ( R d ) to denote the Kato class and the localKato class, respectively [AS, S2]. We also introduce the corresponding infinite-volume self-adjoint Schr¨odinger operators H := − (∆ / V and H := H + V on L ( R d ).The self-adjoint operators H ( L )0 and H ( L )1 have compact resolvents and, there-fore, discrete spectrum. For a given energy E ∈ R , let N ( L )0 ( E ), resp. N ( L )1 ( E ),denote the number of eigenvalues, including multiplicity, for H ( L )0 , resp. H ( L )1 ,less than or equal to E . These are both monotone increasing functions of theenergy E . We define the relative eigenvalue counting function by E ξ L ( E ) ≡ ξ ( E ; H ( L )1 , H ( L )0 ) := N ( L )0 ( E ) − N ( L )1 ( E ) > E ∈ R . It is known that this function is equal to the (more generallydefined) spectral shift function for the pair ( H ( L )1 , H ( L )0 ), see e.g. [Y, BiY] or(5.1) in the Appendix. A basic question is the pointwise boundedness withrespect to the energy of the SSF ξ L as L → ∞ .The main result of this note is Theorem 1.3 which states that the Ces`aromean of subsequences of ξ L is bounded from above by the infinite-volume SSF ξ Lebesgue-almost everywhere. Here, the infinite-volume SSF ξ for the pair( H , H ) is defined in terms of the invariance principle and Kre˘ın’s trace identity,see Remark 5.1 in the Appendix. Theorem 1.3 relies on an abtract result ofKoml´os [Ko] and vague convergence of the measures ξ L ( E ) d E to the measure ξ ( E ) d E , where d E denotes the Lebesgue measure on R . As a by-product ofour analysis we obtain a short proof of the Birman–Solomyak formula for theSSF of Schr¨odinger operators that seems to be new.The main motivation of this note are the two papers of W. Kirsch [K1,K2], who considered the (un-)boundedness of the SSF ξ L as L → ∞ . Weyl’slaw indicates that the leading behavior of each eigenvalue counting functionin (1.1) is the same and proportional to the volume L d . Since the support of V is compact and independent of L , one might think that all L -dependencein (1.1) cancels out and that the SSF remains bounded as L → ∞ . For thecorresponding discrete problem in ℓ ( Z d ), this is indeed true as can be seenfrom a finite-rank perturbation argument. For the continuum problem, however,which we consider here, Kirsch showed that this intuition is wrong in dimensions d > V ≡
0. (In d = 1 the finite-rank perturbation argument is alsoapplicable in the continuum.) Theorem 1.1 ([K1]) . Let d ∈ N \{ } and assume in addition to ( ⋆ ) that V = 0 and V ∈ L ∞ ( R d ) . Then, for any E > , we have sup L> ξ L ( E ) = ∞ . (1.2) Furthermore, there is a countable, dense set of energies
E ⊂ [0 , ∞ [ so that forany E ∈ E , we have sup L ∈ N ξ L ( E ) = ∞ . (1.3) HE SPECTRAL SHIFT FUNCTION 3
The same is true for perturbations by boundary conditions: as a corollary ofTheorem 1.1, Kirsch [K1] considered the Dirichlet Laplacian on Λ L with an addi-tional Dirichlet wall along the boundary ∂ Λ ℓ of an arbitrary fixed cube Λ ℓ ⊂ Λ L .He compared the eigenvalue counting function N ( L ) D,ℓ for this operator, which isa direct sum of two Dirichlet Laplacians, to the one obtained by placing Neu-mann boundary conditions along ∂ Λ ℓ . He concluded that N ( L ) N,ℓ ( E ) − N ( L ) D,ℓ ( E )has an infinite supremum over L , in the same way as in Theorem 1.1. Weremark that, similarly, this effect also shows up when comparing N ( L ) D,ℓ to theeigenvalue counting function of the Dirichlet Laplacian on Λ L .In the general case where V is not identically zero, much less is known.In fact, Kirsch’s proof [K1] of Theorem 1.1 uses the high degeneracy of theeigenvalues of − ∆ L to deduce the claimed divergence of the SSF. In general,the perturbation V removes this degeneracy.In [K2], Kirsch aimed at a complementary statement to Theorem 1.1, askingfor finiteness of the SSF for energies outside the bad countable set in (1.3). Theresult he got, however, requires V to become smaller and smaller in norm when L tends to infinity. Theorem 1.2 ([K2]) . Let d ∈ N and assume in addition to ( ⋆ ) that V , V ∈ L ∞ ( R d ) . Define V L := L − k V for some arbitrary k > d + 1 and H ( L )1 := H ( L )0 + V L . Then, there is a set S ⊂ R of full Lebesgue measure such that forevery E ∈ S , we have lim L →∞ ξ L ( E ) = 0 . (1.4) for the SSF ξ L of the pair ( H ( L )1 , H ( L )0 ) . In our proof of a lower bound on the density of states for random Schr¨odingeroperators [HM], we were also led to consider the question of the boundedness ofthe finite-volume SSF. In the following theorem we prove an almost-everywhereupper bound on the Ces`aro mean of subsequences of ξ L . Our result naturallycomplements Kirsch’s Theorem 1.1, which is commonly cited as an example ofa pathological behavior of the SSF. Theorem 1.3.
Let d ∈ N and assume ( ⋆ ) . Then, for every sequence oflengths ( L j ) j ∈ N ⊂ ]0 , ∞ [ with lim j →∞ L j = ∞ there exists a subsequence ( j i ) i ∈ N ⊂ N with lim i →∞ j i = ∞ such that for every subsequence ( i k ) k ∈ N ⊂ N with lim k →∞ i k = ∞ we have lim K →∞ K K X k =1 ξ e L k ( E ) ξ ( E ) (1.5) for Lebesgue-almost all E ∈ R . Here we have set e L k := L j ik for all k ∈ N . The assumption that V has compact support could be relaxed. There havebeen many works on the boundedness of finite-volume spectral shift functionsfor Schr¨odinger operators. Hundertmark, Killip, Nakamura, Stollmann andVeseli´c [HuKNSV] have obtained a bound on R R d E ξ L ( E ) f ( E ) for bounded,compactly supported functions f . Kostrykin and Schrader [KosS, Ex. 4.2] men-tion that their methods imply pointwise boundedness of the sequence of Laplace P. D. HISLOP AND P. M ¨ULLER transforms (cid:0)e ξ L ( t ) (cid:1) L> for every fixed t >
0. Combes, Hislop and Nakamura[CHN] proved an L p -bound on the SSF for pairs of operators ( A, B ) for which C = B − A is in the Schatten-von Neumann trace ideal I /p , with 1 p < ∞ .This was improved by Hundertmark and Simon [HuS] who obtained an opti-mal bound on the L p -norm of the SSF. Sobolev [So, Sect. 4] showed continuityof the infinite-volume SSF ξ for pairs of Schr¨odinger operators with V = 0and proved a pointwise bound on ξ ( E ) for sufficiently large energies (there aremore general results in [So, Sect. 4] in an abstract setting.) For the case ofrandom Schr¨odinger operators on L ( R d ), it is known that the expectation ofthe finite-volume SSF is pointwise bounded [CHK1, CHK2].The proof of Theorem 1.3 is deferred to Section 3. It relies on a deep resultof Koml´os [Ko, Thm. 1a] for L -bounded sequences. We infer this conditionfrom the non-negativity of ξ L and from vague convergence of the finite-volumeSSF towards the infinite-volume SSF. The latter is the content of Theorem 1.4.
Let d ∈ N and assume ( ⋆ ) . Then, we have lim L →∞ Z R d E ξ L ( E ) f ( E ) = Z R d E ξ ( E ) f ( E ) (1.6) for every continuous and compactly supported function f ∈ C c ( R ) and for everyindicator function f = χ I of some interval I ⊂ R . In particular, for Lebesgue-almost all E ∈ R we have lim δ ↓ lim L →∞ δ Z E + δE d E ′ ξ L ( E ′ ) = ξ ( E ) . (1.7) Remarks . (i) Geisler, Kostrykin, and Schrader [GKS, Thm. 3.3] provedthat the distribution functions ζ L ( E ) := R E −∞ d E ξ L ( E ) of the finite-volumespectral shift measures ξ L ( E ) d E converge to the distribution function ζ ( E ) ofthe infinite-volume spectral shift measure in the case d = 3 and a real-valuedmeasurable potential V ∈ ℓ (L ), the Birman–Solomyak space. In light of[HupLMW, Prop 4.3], this proves vague convergence of the finite-volume spec-tral shift measures. The proof in [GKS] uses Weyl-type high-energy asymptoticsof the SSF (see [GKS, Lemma 2.4]) that are not necessary for our proof.(ii) Kirsch’s result (1.2) shows that one cannot get rid of the energy smooth-ing in (1.7), that is, the limits δ ↓ L → ∞ must not be interchanged. Thebest one could hope for is convergence Lebesgue-almost everywhere of ( ξ L j ) j ∈ N for sequences of diverging lengths. Theorem 1.3 is a partial result in this direc-tion based.(iii) For the sake of concreteness, we mention an example of a toy family( ζ L ) L> of functions R ∋ E ζ L ( E ) := (cid:26) L, E ∈ { Q + [ L ] } , otherwise (1.8)which captures the properties that are known for the family of spectral shiftfunctions ( ξ L ) L> . In (1.8) we have written [ L ] := L mod 1 for the fractionalpart of L in [0 , L →∞ ζ L ( E ) does not exist for any E ∈ R andthe corresponding suprema over L are infinite as in (1.2) and (1.3). The limits(1.6), (1.7) are zero for the toy family. Note that, in addition, the sequence HE SPECTRAL SHIFT FUNCTION 5 of functions ( ζ L j ) j ∈ N converges to zero Lebesgue-almost everywhere for everydiverging sequence ( L j ) j ∈ N of lengths.The Birman–Solomyak formula [BiS, S3] is an important identity relating theSSF to the perturbation potential V . We state it in the next theorem and givea short proof of it in Section 4, valid under our assumptions ( ⋆ ). Even thoughour proof of Theorem 1.4 does not rely upon the Birman–Solomyak formula,they are both related in spirit and are based on the Feynman–Kac formula. Let χ B denote the indicator function of the set B ⊂ R . Theorem 1.6.
Let H and H be as above with potentials V and V satisfying ( ⋆ ) , and let H λ := H + λV on L ( R d ) for λ ∈ [0 , . Then the (infinite-volume)SSF ξ for the pair ( H , H ) satisfies the Birman–Solomyak formula Z B d E ξ ( E ) = Z d λ tr [ V / χ B ( H λ ) V / ] (1.9) for every Borel set B ⊂ R .Remarks . (i) We allow both sides of (1.9) to be + ∞ . If sup B < ∞ ,then tr [ V / χ B ( H λ ) V / ] = Z R d d x V ( x ) χ B ( H λ )( x, x ) < ∞ (1.10)by [Br, Cor. 4.4], the continuity of the integral kernel of the spectral projection(see [AS, Prop. 4.3] or [S2, Thm. B.7.1(d)]) and since V ∈ K ( R d ) ⊆ L , loc ( R d )has compact support. Moreover, this trace is uniformly bounded in λ ∈ [0 , ξ ( E ) d E in Theorem 1.4. Using (1.9), Eq. (1.6) readslim L →∞ Z R d E ξ L ( E ) f ( E ) = Z d λ tr [ V / f ( H λ ) V / ] . (1.11)(iii) Simon remarks that his more general Birman–Solomyak formula [S3,Thm. 4] includes the case of Schr¨odinger operators with slightly different con-ditions on the potentials than ours. For example, one may take V and V to beuniformly Kato class and V > ℓ ( L ).2. Proof of Theorem 1.4
In this section we prove vague convergence of the finite-volume SSF in themacroscopic limit. Our approach is very close to [GKS] but does not requireknowledge of a Weyl asymptotics for high energies. Let
L, t >
0. The standardFeynman–Kac representation [S1] of the heat kernel gives e ξ L ( t ) := Z R d E e − tE ξ L ( E )= 1 t πt ) d/ Z Λ L d x E ,tx,x h χ t Λ L ( b ) e − R t d s V ( b ( s )) (cid:16) − e − R t d s V ( b ( s )) (cid:17)i . (2.1)Here, E ,tx,y denotes the normalized expectation over all Brownian bridge paths b starting at x ∈ R d at time s = 0 and ending at y ∈ R d at time s = t . The P. D. HISLOP AND P. M ¨ULLER
Dirichlet boundary condition is taken into account by the cut-off functional χ t Λ L ( b ), which is equal to one if b ( s ) ∈ Λ L for all s ∈ [0 , t ], and zero otherwise.First, we rewrite the term in parentheses in (2.1) as an integral1 − e − R t d s V ( b ( s )) = Z d λ (cid:18)Z t d s ′ V (cid:0) b ( s ′ ) (cid:1)(cid:19) e − λ R t d s V ( b ( s )) (2.2)over a parameter λ ∈ [0 , b ( s ) to b ( s ) + x and use Fubini’s Theorem so that e ξ L ( t ) = 1 t πt ) d/ Z d λ E ,t , (cid:2) F L ( λ, t ; b ) (cid:3) , (2.3)where F L ( λ, t ; b ) := Z Λ L d x χ t Λ L ( b + x ) (cid:18)Z t d s ′ V (cid:0) b ( s ′ ) + x (cid:1)(cid:19) e − R t d s U λ ( b ( s )+ x ) (2.4)and U λ := V + λV . Clearly, F L ( λ, t ; b ) > L forevery Brownian bridge path b , every λ ∈ [0 ,
1] and every t >
0. Therefore, theMonotone Convergence Theorem gives the pointwise limitlim L →∞ e ξ L ( t ) = 1(2 πt ) d/ Z d λ Z R d d x Z t d st E ,tx,x (cid:2) V (cid:0) b ( s ) (cid:1) U t ( b ) (cid:3) (2.5)for all t >
0, where we have introduced the Brownian functional U t ( b ) := exp (cid:26) − Z t d s U λ (cid:0) b ( s ) (cid:1)(cid:27) (2.6)and used Fubini’s Theorem. (We will see shortly that the limit (2.5) is finite,which, a posteriori, will justify the final interchange of integrations.)Now we show that the limit (2.5) is equal to the two-sided Laplace transformof the infinite-volume SSF. It is well-defined for t >
0, see Remark 5.1, andgiven by e ξ ( t ) := Z R d E e − tE ξ ( E ; H , H ) = − Z R d E e − tE ξ (cid:0) e − tE ; e − tH , e − tH (cid:1) = − t Z ∞ d η ξ (cid:0) η ; e − tH , e − tH (cid:1) = 1 t Z R d η ξ (cid:0) η ; e − tH , e − tH (cid:1) = 1 t tr (cid:0) e − tH − e − tH (cid:1) . (2.7)Here we have used the definition of the SSF in Remark 5.1, and the last equalityfollows from Kre˘ın’s trace formula (5.1). The semigroup difference in the lastline of (2.7) is trace class, cf. Remark 5.1, and possesses a continuous integralkernel. Thus, [Br, Thm. 3.1] justifies the evaluation of the trace by an integralover the diagonal of the kernel so that e ξ ( t ) = 1 t πt ) d/ Z R d d x E ,tx,x h e − R t d s V ( b ( s )) (cid:16) − e − R t d s V ( b ( s )) (cid:17)i = 1(2 πt ) d/ Z d λ Z R d d x Z t d st E ,tx,x (cid:2) V (cid:0) b ( s ) (cid:1) U t ( b ) (cid:3) , (2.8) HE SPECTRAL SHIFT FUNCTION 7 where we have used (2.2) and Fubini’s Theorem. We infer that e ξ ( t ) = lim L →∞ ξ L ( t ) (2.9)for every t >
0. In particular, the limit (2.5) is seen to be finite (as wasused earlier). So the claim (1.6) follows from [F] for f ∈ C c ( R ). But vagueconvergence of a sequence of (unbounded) measures, which are tight at −∞ ,implies pointwise convergence of the corresponding distribution functions atpoints of continuity of the limit, see e.g. [B] or [HupLMW, Prop 4.3]. Thus, (1.6)also holds if f is an indicator function of some interval in R . The statement (1.7)is then a consequence of Lebesgue’s Differentiation Theorem. This completesthe proof of Theorem 1.4.3. Proof of Theorem 1.3
Since the SSF ξ L is non-negative for every L >
0, Theorem 1.4 implies thatfor every sequence ( L n ) n ∈ N of lengths, which is divergent to + ∞ , we havesup n ∈ N Z j − j d E ξ L n ( E ) < ∞ (3.1)for every fixed j ∈ N , that is the sequence is norm bounded in L ([ − j, j ]). In-terpreting ( ξ L n ) n ∈ N as a sequence of uniformly distributed random variables on[ − j, j ], Koml´os’ Theorem [Ko, Thm. 1a] ensures the existence of a subsequence( L n ( j ) ν ) ν ∈ N of lengths and of a function ψ j ∈ L ([ − j, j ]) such that for everyfurther subsequence e L ( j ) k := L n ( j ) νk , k ∈ N , the Ces`aro limitlim K →∞ K K X k =1 ξ e L ( j ) k ( E ) = ψ j ( E ) (3.2)exists for Lebesgue-a.e. E ∈ [ − j, j ]. Here we can assume without restrictionthat ( n ( j +1) ν ) ν ∈ N is a subsequence of ( n ( j ) ν ) ν ∈ N for every j ∈ N . Below we showthat ψ j ξ (3.3)holds Lebesgue-almost everywhere on [ − j, j ] for every j ∈ N . Therefore, givenany subsequence e L k := L n ( νk ) νk , k ∈ N , of the sequence ( L n ( ν ) ν ) ν ∈ N , we get theasserted inequality lim K →∞ K K X k =1 ξ e L k ( E ) ξ ( E ) (3.4)for Lebesgue-a.e. E ∈ R .It remains to establish (3.3) for all j ∈ N . So fix j ∈ N and let f ∈ C ([ − j, j ])arbitrary, subject to f > f ( − j ) = 0 = f ( j ). Then, the trivial extension F of f to R belongs to the positive cone of C c ( R ). We conclude from (3.2) and P. D. HISLOP AND P. M ¨ULLER
Fatou’s Lemma that Z j − j d E f ( E ) ψ j ( E ) = Z R d E F ( E ) (cid:18) lim K →∞ K K X k =1 ξ e L k ( E ) (cid:19) Z R d E F ( E ) ξ ( E ) + lim inf K →∞ K K X k =1 I ( k ) , (3.5)where I ( k ) := R R d E F ( E ) [ ξ e L k ( E ) − ξ ( E )]. Now, the vague convergence ofTheorem 1.4 guarantees that for every ε > k ε ∈ N such that forall k > k ε we have |I ( k ) | ε . This implies (cid:12)(cid:12)(cid:12)(cid:12) lim inf K →∞ K K X k =1 I ( k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) lim inf K →∞ K K X k = k ε I ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ε. (3.6)Since ε > Z j − j d E f ( E ) [ ξ ( E ) − ψ j ( E )] > f ∈ C ([ − j, j ]) that vanishes at the boundary points. Butthis yields (3.3).4. Proof of the Birman–Soloymak formula
In this section we present a simple Feynman–Kac based proof of the Birman–Solomyak formula (1.9). Let us write µ ( B ) for the right-hand side of (1.9),which defines a Borel measure µ on R . We deduce from the spectral theoremand monotone convergence that for every t > e µ ( t ) := R R d µ ( E ) e − tE is given by e µ ( t ) = Z d λ tr (cid:2) V / e − tH λ V / (cid:3) = Z d λ Z R d d x V ( x ) e − tH λ ( x, x )= Z d λ Z R d d x V ( x ) E ,tx,x (cid:2) U t ( b ) (cid:3) (2 πt ) d/ . (4.1)Finiteness of e µ ( t ) for t > V ∈ K ( R d ) ⊆ L , loc ( R d ) has compact support.The functional U t was defined in (2.6).Recall that the probability density ρ ,tx,x ( s ; y ), that the paths of the Brownianbridge satisfy b ( s ) = y ∈ R d for some s ∈ ]0 , t [, is given by ρ ,tx,x ( s ; y ) = (2 πt ) d/ e − ( x − y ) / (2 s ) (2 πs ) d/ e − ( x − y ) / [2( t − s )] [2 π ( t − s )] d/ . (4.2)The Markov property then amounts to the identity E ,tx,x (cid:2) U t ( b ) (cid:3) = Z R d d y ρ ,tx,x ( t − s ; y ) E ,tx,x (cid:2) U t ( b ) (cid:12)(cid:12) b ( t − s ) = y (cid:3) = Z R d d y ρ ,tx,x ( t − s ; y ) E ,t − sx,y (cid:2) U t − s ( b ) (cid:3) E ,sy,x (cid:2) U s ( b ) (cid:3) (4.3) HE SPECTRAL SHIFT FUNCTION 9 for every s ∈ ]0 , t [. Hence, Z R d d x V ( x ) E ,tx,x (cid:2) U t ( b ) (cid:3) = Z t d st Z R d d y Z R d d x ρ ,tx,x ( t − s ; y ) E ,sy,x (cid:2) V (cid:0) b ( s ) (cid:1) U s ( b ) (cid:3) E ,t − sx,y (cid:2) U t − s ( b ) (cid:3) = Z t d st Z R d d y E ,ty,y (cid:2) V (cid:0) b ( s ) (cid:1) U t ( b ) (cid:3) , (4.4)where the second equality relies on ρ ,tx,x ( t − s ; y ) = ρ ,ty,y ( s ; x ) and, again, theMarkov property. A comparison of (4.1) and (4.4) with (2.8) reveals that e µ ( t ) = e ξ ( t ) for all t >
0. Hence, (1.9) follows from [F].5.
Appendix: Basics about the SSF
For the convenience of the reader, we collect some facts related to the SSFin this appendix, see e.g. [Y]. First, we are concerned with its definition in amore general setting. If A , A are self-adjoint operators on a Hilbert space H and if A − A is trace class, then [Y, Thm. 8.3.3 and following remarks] f ( A ) − f ( A ) is trace class for every f ∈ C ∞ c ( R ) and the SSF ξ ≡ ξ ( ··· ; A , , A )of the pair ( A , A ) is uniquely defined up to an additive constant by Kre˘ın’strace formula tr [ f ( A ) − f ( A )] = Z R d E f ′ ( E ) ξ ( E ) . (5.1)The constant can be chosen such that ξ ∈ L ( R ). In this case, we have thebound k ξ k k A − A k tr in terms of the trace norm. We note that thebehavior of f outside the union of the spectra of A and A is irrelevant forboth sides of (5.1) and that (5.1) does also hold for f being the identity.This definition of the spectral shift function ξ ( ··· ; A , , A ) can be extendedto a pair of self-adjoint operators ( A , A ) for which it is only known that ϕ ( A ) − ϕ ( A ) is trace class for some function ϕ ∈ C (Ω), where Ω is a (possiblyinfinite) interval containing the union of the spectra of A and A and ϕ isbounded and strictly monotone on Ω. In this case we set ξ ( E ; A , A ) := sign (cid:0) ϕ ′ ( E ) (cid:1) ξ (cid:0) ϕ ( E ); ϕ ( A ) , ϕ ( A ) (cid:1) (5.2)for all E ∈ Ω. The SSF on the right-hand side of (5.2) is determined by (5.1)and it is independent of the choice of ϕ within the allowed class of functions(invariance principle) [Y, Sect. 8.11]. Moreover, we have the estimate Z Ω d E | ξ ( E ; A , A ) | | ϕ ′ ( E ) | k ϕ ( A ) − ϕ ( A ) k tr . (5.3)Now we return to the situation of Schr¨odinger operators as in the main text. Remark . Let d ∈ N and assume ( ⋆ ). Then, e − tH − e − tH is trace class forevery t >
0, see e.g. [HuKNSV, Remark after Thm. 1], and the SSF for thepair ( H , H ) is defined by (5.2) with ϕ ( E ) := e − tE for E ∈ R . It follows that ξ ( ··· , H , H ) ∈ L ( R ), the integral corresponding to (5.3) is finite, and that itsLaplace transform e ξ ( t ), see (2.7), exists for every t > Acknowledgement.
The authors thank Fran¸cois Germinet for repeated kindhospitality at the Universit´e de Cergy-Pontoise, France.
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Department of Mathematics, University of Kentucky, Lexington, Kentucky40506-0027, USA
E-mail address : [email protected] Mathematisches Institut, Ludwig-Maximilians-Universit¨at, Theresienstraße39, 80333 M¨unchen, Germany
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