Topological Levinson's theorem for inverse square potentials: complex, infinite, but not exceptional
aa r X i v : . [ m a t h . SP ] O c t Topological Levinson’s theorem for inverse squarepotentials: complex, infinite, but not exceptional
H. Inoue, S. Richard ∗ Graduate school of mathematics, Nagoya University, Chikusa-ku,Nagoya 464-8602, Japan
E-mails: [email protected], [email protected]
Abstract
In this review paper we carry on our investigations on Schr¨odinger oper-ators with inverse square potentials on the half-line. Depending on severalparameters, such operators possess either a finite number of complex eigen-values, or an infinite one, but also some spectral singularities embeddedin the continuous spectrum (exceptional situations). The spectral and thescattering theory for these operators is recalled, and new results for the ex-ceptional cases are provided. Some index theorems in scattering theory arealso developed, and explanations why these results can not be extended tothe exceptional cases are provided.
Keywords:
Scattering theory, index theorem, spectral singularity, Fredholm,semi-Fredholm
Levinson’s theorem is a relation between the number of bound states of a quantummechanical system and an expression related to the scattering part of that sys-tem. It was originally established by N. Levinson in [6] for Schr¨odinger operatorswith a spherically symmetric potential, and has then been developed by numerousresearchers on a purely analytical basis. About 10 years ago, it has been shownthat this relation can be interpreted as an index theorem in scattering theory, andthe results of these investigations have been summarized in the review paper [11].More recently, a scattering system involving several parameters has been exhibitedin [3] and this system has been at the root of several extensions of Levinson’s the-orem: In [8] it has been shown that complex eigenvalues can also be counted, and ∗ Supported by the grant
Topological invariants through scattering theory and noncommutativegeometry from Nagoya University, and on leave of absence from Univ. Lyon, Universit´e ClaudeBernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918,F-69622 Villeurbanne cedex, France
1n [5] a first attempt for dealing with an infinite number of eigenvalues has beenintroduced. However, some of the operators exhibited in [3] were not used for theseextensions, and our aim in the present paper is to complete the investigations forthe entire family.Before entering into the details of our investigations, let us immediately men-tion that part of our aim has been unsuccessful. Indeed, for a reduced family ofoperators, we end up with wave operators which are either unbounded or not Fred-holm. In such a situation, computing their Fredholm index is either much moreinvolved or simply not possible. Nevertheless, we provide an exhaustive pictureof the situation and describe the limitations of our approach. We hope that ourpresentation will motivate further investigations for the trickiest cases.Let us now be more precise on the model and on the results, see also Section 2for more details on the model. The initial system consists in a family of Schr¨odingeroperators of the form − ∂ r + (cid:0) m − (cid:1) r on the half-line R + . The parameter m ∈ C with ℜ ( m ) > − m = 0 an additional parameter κ ∈ C is used for defining the boundary conditionat r = 0, while for m = 0 another family of operators indexed by a boundaryparameter ν is defined. The study of the corresponding families of closed operators H m,κ and H ν in L ( R + ) has been initiated and extensively performed in [3].Among all operators H m,κ and H ν only a few are self-adjoint. They are exhib-ited in Lemma 2.2. In the large complementary family, some pairs of parameters( m, κ ) and some parameters ν are called exceptional if they satisfy a prescribedcondition provided in Definition 3.1. As shown in Remark 3.8 the correspondingoperators H m,κ or H ν possess spectral singularities in the continuous spectrum.Around these singularities the spectral and the scattering properties of these op-erators are less obvious, and for that reason these operators were not consideredin [3]. Here, we shall consider all the operators, and provide as much informationas possible even in the exceptional situations.The spectral theory of the operators H m,κ and H ν is provided in Section 3.The number of eigenvalues of these operators can be finite or infinite, dependingon the parameters. For the exceptional operators, it is shown in particular thateven though there is no eigenvalue embedded in the continuous spectrum, it ispossible to construct a family of operators of the same type (but non-exceptional)having complex eigenvalues converging to a prescribed value in R + , see Lemma3.3. These convergences take place either in C + or in C − depending on the choiceof the initial exceptional parameters.The next spectral result corresponds to a limiting absorption principle. Fornon-exceptional operators this limiting process takes place from below and fromabove the real axis in C , but in the exceptional cases some restrictions appear.More precisely, at the spectral singularity the limiting absorption principle holdsonly on one side of the real axis, the side free of possible accumulations of com-plex eigenvalues. These results are gathered in Propositions 3.5 and 3.6. Let usalso note that even if most of the exceptional operators have only one spectralsingularity, some have two spectral singularities (with corresponding limiting ab-sorption principles in two different half-planes) and some have an infinite numberof spectral singularities, converging both to 0 and to + ∞ .2he scattering theory for the pairs of operators ( H m,κ , H D ) or ( H ν , H D ) isstudied in Section 4. The reference operator H D corresponds to the DirichletLaplacian on R + (and is equal to H , ). Following the approach of [3] we start byconstructing the generalized Hankel transformations F ∓ m,κ and F ν ∓ , and definethe wave operators in terms of these transformations. Various representations areprovided for these operators, but here again a special attention has to be paidto the exceptional cases. Indeed, for them either one or sometimes both waveoperators are not bounded. A list of all unbounded wave operators is provided atthe end of Section 4.In the last section we provide some index theorems in scattering theory. Thispart contains new information but the framework corresponds to the one whichalready appeared in [8] and to part of the one used in [5]. The first step consistsin providing a representation of the wave operators in the usual setting of pseudo-differential operators. Since this new representation is implemented by a unitarytransformation, the bounded wave operators remain bounded, and the unboundedones remain unbounded ! However, this representation is convenient for the in-troduction of some C ∗ -algebras containing pseudo-differential operators of order 0and with coefficients which are either asymptotically constant or periodic. Thesealgebras contain all bounded wave operators, as stated in Proposition 5.1.Once in this C ∗ -algebraic framework, the way for index theorems is alreadypaved and rather well understood. Indeed, by looking at some ideals in thesealgebras and by considering the quotient algebras, one ends up automatically withan index map which corresponds to our topological version of Levinson’s theorem.For the model under consideration and depending on the parameters, one obtainseither an index theorem for Fredholm operator or a so-called Atiyah’s L -indextheorem [1]. These results are presented in Theorem 5.3 and 5.5. Note that theFredholm case has already been considered for several models in [11], and besidethe usual contribution due to the scattering operator, two additional contributionsare possible. In the original representation they correspond to corrections at 0-energy and at energy equal to + ∞ . Note also that in the present setting weare dealing with arbitrary complex eigenvalues while in reference [11] only realeigenvalues were considered. On the other hand, when the number of bound statesis infinite, no such correction appears, and the new Levinson’s theorem correspondsto an equality between the winding number computed over one period for thescattering operator, and a suitable trace in the Floquet-Bloch representation ofthe projection on the bound states of H m,κ . This situation coincides with a specialinstance of the results obtained in the seminal paper [1] where an index theoremis provided for elliptic operators on a non-compact manifold which are invariantunder the action of a discrete group. The decomposition with respect to the groupcorresponds in our setting to the Floquet-Bloch decomposition.Unfortunately, in Section 5 about index theorems the exceptional cases are nomore considered. Indeed, as already mentioned some of the corresponding waveoperators are unbounded, and therefore can not easily be associated to any C ∗ -algebra. For the remaining wave operators still belonging to some C ∗ -algebras,their principal symbol are not boundedly invertible. As a consequence, theseoperators are not Fredholm, and their analytical index can not be defined. It is3uite unfortunate that the presence of a spectral singularity prevented us fromdefining any index theorem for the corresponding wave operators. Note that arelated result about the non-completeness of the wave operators in the presence ofspectral singularity has also been recently obtained in [4]. Some relations betweenspectral singularities and scattering theory have also been exhibited in [7, 12],see also references therein. We hope that future investigations will provide newinsights about these exceptional situations in our algebraic framework. In this section we introduce the model used for our investigations. This materialis borrowed from [3] to which we refer for more explanations and for the proofs.Note that [5, 8] also contain partial information of this model.For any m ∈ C we consider the differential expression L m := − ∂ r + (cid:16) m − (cid:17) r acting on distributions on R + . The maximal operator associated with L m in L ( R + ) is defined by D ( L max m ) = { f ∈ L ( R + ) | L m f ∈ L ( R + ) } , and the minimaloperator L min m is defined as the closure of the restriction of L m to C ∞ c ( R + ), where C ∞ c ( R + ) denotes the set of compactly supported smooth functions on R + . Then,the equality ( L min m ) ∗ = L max¯ m holds for any m ∈ C , and L min m = L max m if |ℜ ( m ) | ≥ L min m ( L max m if |ℜ ( m ) | <
1. In the latter situation D ( L min m ) is a closedsubspace of codimension 2 of D ( L max m ), and for any f ∈ D ( L max m ) there exist a, b ∈ C such that f ( r ) − ar / − m − br / m ∈ D ( L min m ) around 0 if m = 0 ,f ( r ) − ar / ln( r ) − br / ∈ D ( L min0 ) around 0 . Here, the expression g ( r ) ∈ D ( L min m ) around ζ ∈ C ∞ c (cid:0) [0 , ∞ ) (cid:1) with ζ = 1 around 0 such that gζ ∈ D ( L min m ). In addition, thebehavior of any function g ∈ D ( L min m ) is known, namely g ∈ H ( R + ) and as r → g ( r ) = o (cid:0) r / (cid:1) and g ′ ( r ) = o (cid:0) r / (cid:1) if m = 0 ,g ( r ) = o (cid:0) r / ln( r ) (cid:1) and g ′ ( r ) = o (cid:0) r / ln( r ) (cid:1) if m = 0 . Remark 2.1.
It is worth mentioning that for m = 0 the functions r r ± m are the two linearly independent solutions of the ordinary differential equation L m u = 0 , and that they are square integrable near if |ℜ ( m ) | < . Similarly, thefunctions r r and r r ln( r ) are the two linearly independent solutions ofthe ordinary differential equation L u = 0 . Based on the above observations we construct various closed extensions of theoperator L min m . For simplicity we restrict our attention to m ∈ C with |ℜ ( m ) | < κ ∈ C ∪ {∞} we define a family of closed operators H m,κ : D ( H m,κ ) = (cid:8) f ∈ D ( L max m ) | for some c ∈ C ,f ( r ) − c (cid:0) κr / − m + r / m (cid:1) ∈ D ( L min m ) around 0 (cid:9) , κ = ∞ ; D ( H m, ∞ ) = (cid:8) f ∈ D ( L max m ) | for some c ∈ C ,f ( r ) − cr / − m ∈ D ( L min m ) around 0 (cid:9) . For m = 0, we introduce an additional family of closed operators H ν with ν ∈ C ∪ {∞} : D ( H ν ) = (cid:8) f ∈ D ( L max0 ) | for some c ∈ C ,f ( r ) − c (cid:0) r / ln( r ) + νr / (cid:1) ∈ D ( L min0 ) around 0 (cid:9) , ν = ∞ ; D ( H ∞ ) = (cid:8) f ∈ D ( L max0 ) | for some c ∈ C ,f ( r ) − cr / ∈ D ( L min0 ) around 0 (cid:9) . Let us directly mention a few simple properties of these operators. For any |ℜ ( m ) | < κ ∈ C ∪ {∞} , the equality H m,κ = H − m,κ − holds. Forthat reason, the case κ = ∞ will be disregarded in the following. In addition,the operator H ,κ does not depend on κ , and all these operators coincide with H ∞ (which has already been fully investigated in [2]). For that reason, all resultsabout the case m = 0 will be formulated in terms of the family H ν for ν ∈ C . Ithas also been proved in [3, Prop. 2.3] that for any m ∈ C with |ℜ ( m ) | < κ, ν ∈ C one has( H m,κ ) ∗ = H ¯ m, ¯ κ and ( H ν ) ∗ = H ¯ ν . Based on this, the self-adjoint elements can easily be identified in the two familiesof operators. Indeed, one has:
Lemma 2.2. (i) The operator H m,κ is self-adjoint for m ∈ ( − , and κ ∈ R ,and for m ∈ i R and | κ | = 1 .(ii) The operator H ν is self-adjoint for ν ∈ R . Let us finally note that the operator H , corresponds to the Dirichlet Lapla-cian on R + , which will be denoted by H D . Later on, this operator will play therole of a comparison operator. In this section we start by introducing the definition of an exceptional pair ( m, κ )or of an exceptional parameter ν . We then show how these exceptional situationsshow off in spectral theory, by recalling a few spectral result obtained in [3] andby making some of them slightly more accurate.5n the sequel, we shall use the notations z ln( z ) for the principal value of thelogarithm whose imaginary part lies in the interval ( − π, π ]. On the other hand,Ln( z ) will denote the multivalued logarithm. This means that Ln( z ) = { ln( z ) +2 πi Z } , or equivalently if w satisfies e w = z , then Ln( z ) is the set { w + 2 πi Z } . Wealso introduce for m ∈ C ∗ with |ℜ ( m ) | < κ ∈ C the new parameter ς ≡ ς ( m, κ ) := κ Γ( − m )Γ( m )where Γ denotes the usual Gamma function. Definition 3.1.
A pair ( m, κ ) in C ∗ × C ∗ with |ℜ ( m ) | < is called exceptional if ± π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) . Similarly, a parameter ν ∈ C is called exceptional if ℑ ( ν ) = ± π . Let us immediately stress that ± π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) are two independent condi-tions. Indeed by setting m = m r + im i ∈ C ∗ with m i , m r ∈ R and | m r | < α ∈ ℑ (cid:16) m Ln( ς ) (cid:17) ⇔ α ∈ ℑ (cid:16) m (cid:0) ln( ς ) + 2 πi Z (cid:1)(cid:17) ⇔ α ∈ ℑ (cid:16) m ln( ς ) (cid:17) + 2 π ℜ (cid:16) m (cid:17) Z ⇔ α ∈ ℑ (cid:16) m ln( ς ) (cid:17) + 2 π m r m + m Z . (3.1)Thus, this equation can either be satisfied for α = π or for α = − π , or can besatisfied both for π and − π . Indeed, if m r = 0 both conditions are simultaneouslysatisfied if and only if the following system of equations is satisfied for some z , z ∈ Z with z = z : ( ℑ (cid:0) m ln( ς ) (cid:1) = − π z + z z − z m + m m r = z − z . On the other hand, if m = in for some n ∈ R ∗ then (3.1) corresponds to α = − (cid:0) n ln( | κ | ) (cid:1) .Let us now recall some information about the point spectrum of the operators H m,κ or H ν . Theorem 3.2 (Theorem 5.2 in [3]) . Let m ∈ C with |ℜ ( m ) | < .(i) For m = 0 one has σ p ( H m, ) = ∅ while for κ ∈ C ∗ one has σ p ( H m,κ ) = n − − w | w ∈ m Ln( ς ) and − π < ℑ ( w ) < π o (ii) For any ν ∈ C , σ p ( H ν ) is nonempty if and only if − π < ℑ ( ν ) < π , andthen σ p ( H ν ) = (cid:8) − ν − γ ) (cid:9) where γ denotes the Euler’s constant. σ p ( H m,κ ) depends in a complicated wayon the parameters m and κ . There even exists a pattern of phase transitions , whensome eigenvalues disappear in the continuous spectrum. By looking carefully onthe conditions appearing in the above statement we can see that the exceptionalsituations correspond to the borderline cases, and that the location of the eigen-values are not arbitrary. More precisely if we set C ± := { z ∈ C | ±ℑ ( z ) > } thenone has: Lemma 3.3. (i) Let ( m n ) n ∈ N ⊂ { z ∈ C ∗ | |ℜ ( z ) | < } and ( κ n ) n ∈ N ⊂ C ∗ betwo sequences, set ς n := κ n Γ( − m n )Γ( m n ) , a n := ℜ (cid:16) m n ln( ς n ) (cid:17) , b n := ℑ (cid:16) m n ln( ς n ) (cid:17) and assume that ( a n ) n ∈ N converges to a ∞ ∈ R and that ( b n ) n ∈ N is an in-creasing sequence converging to π . Then, for n large enough there exists λ n ∈ σ p ( H m n ,κ n ) with λ n ∈ C + and λ n → − a ∞ as n → ∞ . If ( b n ) n ∈ N isa decreasing sequence converging to − π , then for n large enough λ n ∈ C − and λ n → − a ∞ as n → ∞ . The value − a ∞ is not an eigenvalue for anyoperator H m,κ .(ii) Let ( a n ) n ∈ N ⊂ R be a sequence converging to a ∞ ∈ R and let ( b n ) n ∈ N ⊂ (0 , π ) be an increasing sequence converging to π . Then, for any n ∈ N and for ν n := a n + ib n one has σ p ( H ν n ) = { λ n } ⊂ C − and λ n → a ∞ − γ ) as n → ∞ . If ( b n ) n ∈ N ⊂ ( − π , is a decreasing sequence converging to − π ,then one has σ p ( H ν n ) = { λ n } ⊂ C + and λ n → a ∞ − γ ) as n → ∞ . Thevalue a ∞ − γ ) is not an eigenvalue for any operator H ν .Proof. The proof simply consists in an application of Theorem 3.2.Let us recall one more result related to eigenvalues. Later on, we shall needthe following characterization of σ p ( H m,κ ), i.e. of the number of eigenvalues of H m,κ . Proposition 3.4 (Proposition 5.3 in [3]) . Let m = m r + im i ∈ C ∗ with | m r | < ,and let κ ∈ C ∗ .(i) If m r = 0 and ln( | κ | ) m i ∈ ( − π, π ) then σ p ( H m,κ ) = ∞ ,(ii) If m r = 0 and ln( | κ | ) m i ( − π, π ) then σ p ( H m,κ ) = 0 ,(iii) If m r = 0 and if N ∈ { , , , . . . } satisfies N < m + m | m r | ≤ N + 1 , then σ p ( H m,κ ) ∈ { N, N + 1 } . Let us now turn to the continuous spectrum for the operators H m,κ and H ν .It has been shown in [3] that [0 , ∞ ) belongs to the spectrum of all these operators.In addition, a limiting absorption principle has been exhibited. Such a resultcorresponds to the existence of a boundary value of the resolvent on (0 , ∞ ) when7onsidered in some weighted Hilbert spaces. For that purpose, we introduce for t > H t and H − t with H t the domain of the operator h R i t of multiplication by the function r
7→ h r i t ≡ (1 + r ) t/ in L ( R + ), and H − t stands for its dual space. We also recall the definition of the Bessel functions fordimension 1 as introduced and motivated in [3], namelythe modified Bessel function for dimension I m ( z ) := r πz I m ( z ) , the MacDonald function for dimension K m ( z ) := r zπ K m ( z ) , the Bessel function for dimension J m ( z ) := r πz J m ( z ) , the Hankel function of the 1st kind for dimension H + m ( z ) := r πz H + m ( z ) , the Hankel function of the 2nd kind for dimension H − m ( z ) := r πz H − m ( z ) , the Neumann function for dimension Y m ( z ) := r πz Y m ( z ) , where I m is the modified Bessel function, K m is the MacDonald function, J m isthe Bessel function, H ± m are the Hankel function of the 1st kind and of the 2ndkind, and Y m is the Neumann function.In the following statements, we recall the limiting absorption principle obtainedin [3] and improve the statement in the exceptional situations. The operators H m,κ and H ν are considered separately, and we set R m,κ ( z ) := ( H m,κ − z ) − and R ν ( z ) := ( H ν − z ) − for their resolvents. We also set when κ = 0Ω ± m,κ := n k ∈ R + | k = 4e −ℜ ( m ln( ς ) ) e π ℑ ( m ) z for any z ∈ Z satisfying ± π = ℑ (cid:0) m (ln( ς ) + 2 πiz ) (cid:1)o . (3.2)As a consequence of the observation made after Definition 3.1 the set Ω ± m,κ isempty if ( m, κ ) is not an exceptional pair, it consists of one single value if ± π ∈ℑ (cid:0) m (ln( ς ) + 2 πi Z ) (cid:1) and m r = 0, but it consists of an infinite set if m r = 0 and ± π = − (cid:0) n ln( | κ | ) (cid:1) . Proposition 3.5.
Let m ∈ C ∗ with |ℜ ( m ) | < , let κ ∈ C , and let k > .(i) If ( m, κ ) is not an exceptional pair, then the boundary values of the resolvent R m,κ ( k ± i
0) := lim ǫ ց R m,κ ( k ± iǫ ) exist in the sense of operators from H t to H − t for any t > , uniformly in k on each compact subset of R + . The kernel of R m,κ ( k ± i is given for < r ≤ s by R m,κ ( k ± i r, s )= ± ik (cid:0) − ς e ∓ iπm (cid:0) k (cid:1) m (cid:1) (cid:16) J m ( kr ) − ς (cid:0) k (cid:1) m J − m ( kr ) (cid:17) H ± m ( ks ) and the same expression with the role of r and s exchanged for < s < r .(ii) If π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but − π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then the statement (i) holds for R m,κ ( k − i , while for R m,κ ( k + i it only holds uniformly in k on eachcompact subset of R + \ Ω + m,κ .(iii) If − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then the statement (i) holds for R m,κ ( k + i , while for R m,κ ( k − i it only holds uniformly in k on eachcompact subset of R + \ Ω − m,κ .(iv) If π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) and − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then the statement (i) holds for R m,κ ( k + i uniformly in k on each compact subset of R + \ Ω + m,κ , and for R m,κ ( k − i uniformly in k on each compact subset of R + \ Ω − m,κ . For the next statement we setΩ ν := (cid:8) k ∈ R + | k = 4e ℜ ( ν ) − γ ) (cid:9) . Proposition 3.6.
Let ν ∈ C , and let k > .(i) If ν is not an exceptional parameter, then the boundary values of the resolvent R ν ( k ± i
0) := lim ǫ ց R ν ( k ± iǫ ) exist in the sense of operators from H t to H − t for any t > , uniformlyin k on each compact subset of R + . The kernel of R ν ( k ± i is given for < r ≤ s by R ν ( k ± i r, s )= ± ik (cid:0) γ + ln (cid:0) k (cid:1) − ν ∓ i π (cid:1) (cid:16)(cid:0) γ + ln (cid:0) k (cid:1) − ν (cid:1) J ( kr ) − π Y ( kr ) (cid:17) H ± ( ks ) . and the same expression with the role of r and s exchanged for < s < r .(ii) If ℑ ( ν ) = π , then the statement (i) holds for R ν ( k + i while for R ν ( k − i it only holds uniformly in k on each compact subset of R + \ Ω ν .(iii) If ℑ ( ν ) = − π , then the statement (i) holds for R ν ( k − i while for R ν ( k + i it only holds uniformly in k on each compact subset of R + \ Ω ν .Proof of Propositions 3.5 & 3.6. In the non exceptional situations, these state-ments already appeared in [3, Thm. 6.1 & Prop. 7.1] while in the exceptionalsituations the statements above are slightly more precise than the corresponding9nes in this reference. In fact, the only difference is that a more careful analysis ofsome numerical prefactors is considered here. Namely, let us consider the followingequivalences:1 − ς e ∓ iπm (cid:0) k (cid:1) m = 0 ⇔ ς = e m ( ± iπ − ln( k ) ) ⇔ m Ln( ς ) ∋ ± iπ − ln (cid:16) k (cid:17) ⇔ ( ± π = ℑ (cid:0) m (ln( ς ) + 2 πiz ± ) (cid:1) k = 4e −ℜ ( m ln( ς ) ) e π ℑ ( m ) z ± whenever such z ± ∈ Z exist. Then, the convergences mentioned in the statementonly hold if the factor (cid:0) − ς e ∓ iπm (cid:0) k (cid:1) m (cid:1) − does not vanish, and the previouscomputation explains the necessary restriction in the exceptional situations.For the second statement, it is sufficient to observe that γ + ln (cid:16) k (cid:17) − ν ∓ i π ⇔ ( ℑ ( ν ) = ∓ π k = 4e ℜ ( ν ) − γ ) and the same argument allows us to conclude.Before turning our attention to scattering theory, let us add two remarks relatedto the above limiting absorption principle: Remark 3.7.
The information provided on the discrete spectrum and on the con-tinuous spectrum are quite consistent. Indeed, let us compare the content of Lemma3.3 with the previous two propositions. If ( m, κ ) is an exceptional pair, the limitingabsorption principle holds without limitation in the half-plane in C where there isno possible accumulation of eigenvalues of some operators in the same family. Onthe other hand, in the half-plane where there is a possible accumulation of eigen-values the limiting absorption principle holds only away from these singular points.A similar observation is also valid for the operator H ν when ν is an exceptionalparameter. Remark 3.8.
The elements in the sets Ω ± m,κ and Ω ν correspond to spectral singu-larities of the operators H m,κ and H ν respectively, see [13, Sec. 2] and [4, Sec. 2.3]for more information on this concept. In our setting, it means that if k o ∈ Ω ± m,κ then the following limits hold uniformly in k on suitable neighborhood of k o , namely lim ǫ ց | k − k o | R m,κ ( k ± iǫ ) exist in the sense of operators from H t to H − t for t > . Similar limits also holdfor k o ∈ Ω ν and for the resolvent of the operator H ν . Note finally that if ( m, κ ) isan exceptional pair and if ℜ ( m ) = 0 , then H m,κ has an infinite number of spectralsingularities. Scattering theory
In this section we review the scattering theory for the operators H m,κ and H ν asdeveloped in [3]. However, exceptional situations were disregarded in this refer-ence, so we also provide new information about the corresponding operators.First of all, let us recall the definition of the Hankel transform. For any m ∈ C with ℜ ( m ) > − F m : C c ( R + ) → L ( R + ) with (cid:0) F m f (cid:1) ( r ) := Z ∞ F m ( r, s ) f ( s )d s and F m ( r, s ) := r π J m ( rs ) . It has been shown in [3, Prop. 4.5] that this map extends continuously to a boundedinvertible operator in L ( R + ), with F − m = F m . Additional information aboutthis operator will be provided later on.Based on this transformation, for any m ∈ C with |ℜ ( m ) | < κ ∈ C let us define the incoming and outgoing Hankel transformations F ∓ m,κ givenby F ∓ m,κ = (cid:16) F m − ς F − m (cid:0) R (cid:1) m (cid:17) e ∓ i π m − ς e ∓ iπm (cid:0) R (cid:1) m . As already seen in the proof of Propositions 3.5, the denominator in the lastfactor vanishes only if ( m, κ ) is an exceptional pair. However, even in this casethe operators F ∓ m,κ are still well defined as unbounded operator on several naturaldomains. For example, this operator is well defined on the set C c (cid:0) R + \ Ω ± m,κ (cid:1) withΩ ± m,κ introduced in (3.2).In order to get a better understanding of these operators, let us provide aslightly modified presentation of them. For that purpose, we define the unitaryand self-adjoint transformation J : L ( R + ) → L ( R + ) by the formula (cid:0) Jf (cid:1) ( r ) = 1 r f (cid:16) r (cid:17) for any f ∈ L ( R + ) and r ∈ R + . We also denote by A the generator of dilationgroup in L ( R + ), namely the generator of the unitary group { U τ } τ ∈ R satisfying[ U τ f ]( r ) = e τ/ f (e τ r ) for any τ ∈ R , f ∈ L ( R + ) and r ∈ R + . Finally weintroduce the bounded and continuous function Ξ m : R → C defined for t ∈ R byΞ m ( t ) := e i ln(2) t Γ( m +1+ it )Γ( m +1 − it ) . (4.1)We can now provide a slightly generalized version of [3, Lem. 6.3] : Lemma 4.1.
For any m ∈ C ∗ with |ℜ ( m ) | < and any κ ∈ C the followingequality holds: F ∓ m,κ = J (cid:16) Ξ m ( A ) − ς Ξ − m ( A ) (cid:0) R (cid:1) m (cid:17) e ∓ i π m − ς e ∓ iπm (cid:0) R (cid:1) m . (4.2)11 f ( m, κ ) is not an exceptional pair, then this equality holds between bounded oper-ators, while if ( m, κ ) is an exceptional pair, then:(i) If π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but − π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then F + m,κ extends to a boundedoperator while the above equality for F − m,κ holds on C c (cid:0) R + \ Ω + m,κ (cid:1) .(ii) If − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then F − m,κ extends to a boundedoperator while the above equality for F + m,κ holds on C c (cid:0) R + \ Ω − m,κ (cid:1) .(iii) If π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) and − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) , the above equality holds for F − m,κ on C c (cid:0) R + \ Ω + m,κ (cid:1) and for F + m,κ on C c (cid:0) R + \ Ω − m,κ (cid:1) . Similarly, for any ν ∈ C we define the incoming and outgoing Hankel transfor-mations F ν ∓ given by the kernels for r, s ∈ R + : F ν ∓ ( r, s ) := r π (cid:16) J ( rs ) ± i π γ + ln (cid:0) s (cid:1) − ν ∓ i π H ± ( rs ) (cid:17) = r π (cid:18) (cid:0) γ + ln (cid:0) s (cid:1) − ν (cid:1) J ( rs ) − π Y ( rs ) γ + ln (cid:0) s (cid:1) − ν ∓ i π (cid:19) . However, a better understanding of these transformations can be obtained withthe subsequent formulas:
Lemma 4.2.
For any ν ∈ C the following alternative description of F ν ∓ hold: F ν ∓ = J Ξ ( A ) (cid:16) γ + ln (cid:0) R (cid:1) − ν − i π tanh (cid:0) π A (cid:1)(cid:17) γ + ln (cid:0) R (cid:1) − ν ∓ i π . (4.3) If ν is not an exceptional parameter, then this equality holds between boundedoperators, while if ν is an exceptional pair, then:(i) If ℑ ( ν ) = π , then F ν − extends to a bounded operator while the above equalityfor F ν +0 holds on C c (cid:0) R + \ Ω ν (cid:1) .(ii) If ℑ ( ν ) = − π , then F ν +0 extends to a bounded operator while the aboveequality for F ν − holds on C c (cid:0) R + \ Ω ν (cid:1) .Proof of Lemmas 4.1 & 4.2. In the non exceptional cases, these statements andtheir proofs already appeared in [3, Lem. 6.3 & Corol. 7.6]. It is then enough toobserve that the same arguments hold in the exceptional cases, when the possiblesingularities of the multiplication operators are taken into account by choosingsuitable domains for these operators.Before introducing the wave operators, let us observe that the operators F ∓ , take a very explicit form. Indeed, as shown in [3, Sec. 4.7] one has F ∓ D ≡ F ∓ , = e ∓ i π Ξ ( − A ) J = e ∓ i π F D (cid:0) F D f (cid:1) ( r ) := r π Z ∞ sin( rs ) f ( s ) d s, f ∈ L ( R + ) . This operator is clearly unitary.We can now introduce the wave operators for the pairs ( H m,κ , H D ) or ( H ν , H D ),where H D ≡ H , denotes the Dirichlet Laplacian on R + . Note that H D for thereference operator is chosen for simplicity, but other choices are possible and leadto interesting phenomena, as emphasized in [5]. In [3] the wave operators aredefined by the formulas W ∓ m,κ ≡ W ∓ ( H m,κ , H D ) := F ∓ m,κ F ± D and W ν ∓ ≡ W ∓ ( H ν , H D ) := F ν ∓ F ± D . However, since some of these operators are unbounded in the exceptional cases,we shall use a unitarily equivalent definition for these operators, namely we shallconsider F ± D W ∓ ( F ± D ) − , or more precisely W ∓ m,κ := F ± D F ∓ m,κ = Ξ ( − A ) (cid:16) Ξ m ( A ) − ς Ξ − m ( A ) (cid:0) R (cid:1) m (cid:17) e ∓ i π ( m − ) − ς e ∓ iπm (cid:0) R (cid:1) m and W ν ∓ := F ± D F ν ∓ = Ξ ( − A )Ξ ( A ) (cid:16) γ + ln (cid:0) R (cid:1) − ν − i π tanh (cid:0) π A (cid:1)(cid:17) e ± i π γ + ln (cid:0) R (cid:1) − ν ∓ i π . As a direct consequence of Lemmas 4.1 and 4.2 most of these operators arebounded and thus well defined on L ( R + ). However, in the exceptional cases someof the operators F ± m,κ are not bounded, and thus were only defined on suitabledomains. For that reason the corresponding wave operators are also only definedon the same domains. For completeness, let us enumerate the operators which areunbounded, and consequently which require a special attention for their definition:(i) If π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but − π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then W − m,κ is unbounded.(ii) If − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then W + m,κ is unbounded.(iii) If π ∈ ℑ (cid:0) m (Ln( ς ) (cid:1) and − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) , then both operators W ± m,κ areunbounded.(iv) If ℑ ( ν ) = π , then W ν +0 is unbounded.(v) If ℑ ( ν ) = − π , then W ν − is unbounded.Except the operators appearing in the above list, all wave operators are bounded.13 emark 4.3. In the previous two lemmas and in the above statement we have beenrather pessimistic, and there is a tiny chance that the situation is slightly betterthan described. Indeed, if one looks carefully at the expressions provided in (4.2) and (4.3) the operator F ∓ m,κ or F ν ∓ consist in a product of two types of operators,namely some functions of A and some multiplication operators (functions of R ).For all parameters m , κ and ν the functions of A are bounded. On the otherhand, depending on the parameters m , κ and ν the multiplication operators areeither bounded or not. In the latter case, the operators W ∓ m,κ or W ν ∓ can bedefined on a natural domain for the unbounded multiplication operators, but it isnot clear if this domain can be extended due to some cancellations with the otherfactors. In the above statements, we took the precautious attitude of not expectingany improvement, and for that reason we mentioned that some wave operators areunbounded. It would certainly be interesting to further investigate in this directionand get a better description of the maximal domain of these operators and of theirrange. So far, our attempts have not been successful. In this section we provide the algebraic framework which leads to a topologicalversion of Levinson’s theorem. This framework for the current model alreadyappeared in [5, 8], but we extend the results presented in these references in threedirections. Indeed, we shall consider systems with arbitrary eigenvalues (complexor real) and in arbitrary number (finite or infinite). In the former reference, onlyreal eigenvalues were considered (which means only self-adjoint operators H m,κ were studied), and in the latter only a finite number of complex eigenvalues wereconsidered. In addition, we also provide an index theorem for the pair ( H ν , H D )which has never been exhibited before.Our first task is to provide a more familiar but unitarily equivalent represen-tation of the wave operators. Indeed, since the operators A and B := ln (cid:0) R (cid:1) in L ( R + ) satisfy the Weyl commutation relation, they are unitarily equivalent tothe operators D = − i∂ x and X in L ( R ). This equivalence is essentially imple-mented by a Mellin transform. Through this transformation the wave operatorsintroduced in the previous section are given by the following expressions: W ∓ m,κ := Ξ ( − D ) (cid:16) Ξ m ( D ) − ς Ξ − m ( D )e mX (cid:17) e ∓ i π ( m − ) − ς e ∓ iπm e mX (5.1)and W ν ∓ := Ξ ( − D )Ξ ( D ) (cid:16) γ + X − ν − i π tanh (cid:0) π D (cid:1)(cid:17) e ± i π γ + X − ν ∓ i π . (5.2)This representation is more familiar since these operators correspond now topseudo-differential operators.Our second task is to observe that these operators are made of functions of D X which have precise properties. For that purpose, let us set when κ = 0Λ ± m,κ := n x ∈ R | x = − (cid:16) ℜ (cid:0) m ln( ς ) (cid:1) − π ℑ (cid:0) m (cid:1) z (cid:17) for any z ∈ Z satisfying ± π = ℑ (cid:0) m (ln( ς ) + 2 πiz ) (cid:1)o and Λ ν := ∅ if ℑ ( ν ) = ± π while if ℑ ( ν ) = ± π Λ ν := (cid:8) x ∈ R | x = ℜ ( ν ) − γ (cid:9) . These sets are the counterparts of Ω ± m,κ and Ω ν in the new representation. Wethen define the functions of two variables: Γ ∓ m,κ : R \ Λ ± m,κ × R → C and Γ ν ∓ : R \ Λ ν × R → C byΓ ∓ m,κ ( x, ξ ) := Ξ ( − ξ ) (cid:16) Ξ m ( ξ ) − ς Ξ − m ( ξ )e mx (cid:17) e ∓ i π ( m − ) − ς e ∓ iπm e mx and Γ ν ∓ ( x, ξ ) := Ξ ( − ξ )Ξ ( ξ ) (cid:16) γ + x − ν − i π tanh (cid:0) π ξ (cid:1)(cid:17) e ± i π γ + x − ν ∓ i π . Formally the following equalities hold: W ∓ m,κ = Γ ∓ m,κ ( X, D ) and W ν ∓ = Γ ν ∓ ( X, D ) , but the only precise meaning is the one provided in (5.1) and (5.2).Let us now introduce the commutative algebra C (cid:0) [ −∞ , ∞ ] (cid:1) of continuous func-tions on R having limits at ±∞ . We then recall from the proof of [3, Thm. 4.10]that for any m, m ′ ∈ C with ℜ ( m ) > − ℜ ( m ′ ) > − ξ Ξ m ( − ξ )Ξ m ′ ( ξ ) belongs to C (cid:0) [ −∞ , ∞ ] (cid:1) and that the following equalities hold:Ξ m ( ∓∞ )Ξ m ′ ( ±∞ ) = e ∓ i π ( m − m ′ ) . We also introduce two non-commutative algebras which are going to nest thewave operators W ∓ m,κ and W ν ∓ . Firstly, we consider the unital C ∗ -subalgebra of B (cid:0) L ( R ) (cid:1) E o = C ∗ (cid:16) a ( D ) b ( X ) | a ∈ C (cid:0) [ −∞ , ∞ ] (cid:1) , b ∈ C (cid:0) [ −∞ , + ∞ ] (cid:1)(cid:17) . Secondly, for any n > C ∗ -subalgebra of B (cid:0) L ( R ) (cid:1) E n := C ∗ (cid:16) a ( D ) b ( X ) | a ∈ C (cid:0) [ −∞ , + ∞ ] (cid:1) , b ∈ C πn ( R ) (cid:17) , where C πn ( R ) denotes the set of all continuous periodic functions on R with period πn . Based on a careful analysis of the functions Γ ∓ m,κ and Γ ν ∓ one easily deducesthe following statement, see [5] and [8] for similar results.15 roposition 5.1. Let m ∈ C ∗ with |ℜ ( m ) | < , and let κ, ν ∈ C . The followingoperators belong to E o :I.1) W ∓ m,κ if ( m, κ ) is not an exceptional pair and ℜ ( m ) = 0 ,I.2) W + m,κ if ℜ ( m ) = 0 and π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but − π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) ,I.3) W − m,κ if ℜ ( m ) = 0 and − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) ,I.4) W ν ∓ if ν is not an exceptional parameter,I.5) W ν − if ℑ ( ν ) = π , and W ν +0 if ℑ ( ν ) = − π .The following operators belong to E | n | :II.1) W ∓ m,κ if ( m, κ ) is not an exceptional pair and m = in for some n ∈ R ∗ ,II.2) W + m,κ if m = in for some n ∈ R ∗ and π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but − π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) ,II.3) W − m,κ if m = in for some n ∈ R ∗ and − π ∈ ℑ (cid:0) m Ln( ς ) (cid:1) but π
6∈ ℑ (cid:0) m Ln( ς ) (cid:1) .In all other cases, the wave operators are not bounded and do no belong to any C ∗ -algebras. Remark 5.2.
When an operator is unbounded, it can not belong to any C ∗ -algebrabut it is still possible that its resolvent belong it. It is thus a natural question tocheck if the unbounded wave operators belong to the C ∗ -algebras introduced above.Since some essential information on these operators are still missing, as alreadymentioned in Remark 4.3, we can not answer this question. Once we know that the wave operators belong to very explicit C ∗ -subalgebra of B (cid:0) L ( R ) (cid:1) , the third task consists in studying these algebras and their structures,and to deduce a suitable C ∗ -algebra framework for deducing index theorems. Thetwo algebras E o and E n will be studied independently, and we shall start with theformer one.The key observation for the analysis of E o is that the ideal of compact operators K R := K (cid:0) L ( R ) (cid:1) corresponds to the C ∗ -algebra generated by products of the form a ( D ) b ( X ) with a, b ∈ C ( R ), with C ( R ) the algebra of continuous functions on R vanishing at ±∞ . Then, one easily infers that E o / K R is isomorphic to C ( (cid:3) ), thealgebra of continuous functions on the boundary (cid:3) of the closed square (cid:4) . Notethat the unital quotient morphism q o : E o → C ( (cid:3) ) is uniquely determined by q o (cid:0) a ( D ) b ( X ) (cid:1) = (cid:0) a ( · ) b ( −∞ ) , a ( −∞ ) b ( · ) , a ( · ) b (+ ∞ ) , a (+ ∞ ) b ( · ) (cid:1) . In fact, the above notation corresponds to an embedding of the algebra C ( (cid:3) ) asa subalgebra of C (cid:0) [ −∞ , + ∞ ] (cid:1) ⊕ C (cid:0) [ −∞ , + ∞ ] (cid:1) ⊕ C (cid:0) [ −∞ , + ∞ ] (cid:1) ⊕ C (cid:0) [ −∞ , + ∞ ] (cid:1) given by elements (Γ , Γ , Γ , Γ ) which coincide at the corresponding end points,that is, Γ ( −∞ ) = Γ ( −∞ ), Γ (+ ∞ ) = Γ ( −∞ ), Γ (+ ∞ ) = Γ (+ ∞ ), andΓ ( −∞ ) = Γ (+ ∞ ). 16ow, whenever the wave operators W ∓ m,κ and W ν ∓ belong to E o , the corre-sponding functions Γ ∓ m,κ or Γ ν ∓ admit a restriction to (cid:3) . More precisely in sucha situation one easily deduces the following expressions for the restriction of Γ − m,κ to (cid:3) : Γ − m,κ ;1 ( ξ ) = ( e i π ( − m ) Ξ ( − ξ )Ξ m ( ξ ) if ℜ ( m ) > , e i π ( + m ) Ξ ( − ξ )Ξ − m ( ξ ) if ℜ ( m ) < , Γ − m,κ ;2 ( x ) = e iπ ( − m ) − ς e + iπm e mx − ς e − iπm e mx , Γ − m,κ ;3 ( ξ ) = ( e i π ( + m ) Ξ ( − ξ )Ξ − m ( ξ ) if ℜ ( m ) > , e i π ( − m ) Ξ ( − ξ )Ξ m ( ξ ) if ℜ ( m ) < , Γ − m,κ ;4 ( x ) = 1 , and in the special case κ = 0 one has Γ − m, ( ξ ) = Γ − m, ( ξ ) = e i π ( − m ) Ξ ( − ξ )Ξ m ( ξ ),Γ − m, ( x ) = e iπ ( − m ) and Γ − m, ( x ) = 1. Similarly, one has for the restriction ofΓ ν − to (cid:3) Γ ν − ( ξ ) = e i π Ξ ( − ξ )Ξ m ( ξ )Γ ν − ( x ) = e i π γ + x − ν + i π γ + x − ν − i π , Γ ν − ( ξ ) = e i π Ξ ( − ξ )Ξ m ( ξ )Γ ν − ( x ) = 1 , Note also that similar expressions for Γ + m,κ and Γ ν +0 can be computed, but theyare not presented for brevity.Before stating our first index theorem with the data mentioned above, twomore information coming from scattering theory are necessary. The first one isrelated to the scattering operator. Let us recall that for a scattering system definedby the pair of operators ( H m,κ , H D ) the scattering operator S m,κ is given by theproduct W − m,κ W − m,κ where means the transpose operator. Note that in theself-adjoint case this definition corresponds to the more usual product W + ∗ m,κ W − m,κ and extends it when the operators are not self-adjoint. The scattering operatoris known to commute with the reference operator, namely H D . In addition, thisoperator is unitarily equivalent to the multiplication operator in L ( R ) definedby the function Γ − m,κ ;2 . By analogy this operator will also be denoted by S m,κ .Clearly, a similar relation exists between the scattering operator S ν for the pair( H ν , H D ) and the multiplication operator in L ( R ) defined by the function Γ ν − ≡S ν . A key observation about these scattering operators is that whenever ( m, κ )or ν are not exceptional the functions Γ − m,κ ;2 or Γ ν − are bounded and boundedlyinvertible on R . In these situations it then follows that the functions Γ − m,κ and Γ ν − defined on (cid:3) are also invertible and boundedly invertible. As a consequence, theirwinding numbers are well defined and will be denoted by Wind. On the other Recall that for a function f defined on a closed curve γ and taking values in C ∗ its winding m, κ ) is not an exceptional pair the kernels of W ∓ m,κ areempty while the cokernels of these operators corresponds to the subspaces spannedby the eigenfunctions associated to the eigenvalues of H m,κ . Similarly, if ν is notan exceptional parameter, the wave operators W ν ∓ have an empty kernel and acokernel equal to the subspace spanned by the eigenfunctions of H ν .With all the information collected so far, the following statement can easily beproved. It relies on the index map associated to the short exact sequence0 → K R → E o → C ( (cid:3) ) → W − m,κ is a lift for the invertible operator Γ − m,κ ∈ C ( (cid:3) ) (andsimilarly W ν − is a lift for the invertible operator Γ ν − ∈ C ( (cid:3) )). The details areprovided in [8]. Note that the following statement applies for the cases I.1) andI.4) of Proposition 5.1. Let us still recall for clarity that the index of a Fredholmoperator corresponds to the difference between the dimension of its kernel and ofits cokernel. This index will be denoted by Index in the sequel. Theorem 5.3 (Topological Levinson’s theorem) . Let m ∈ C ∗ with |ℜ ( m ) | ∈ (0 , ,and let κ, ν ∈ C . If ( m, κ ) and ν are not exceptional parameters, then the followingrelations hold: Wind (cid:2) Γ − m,κ (cid:3) = number of eigenvalues of H m,κ and Wind (cid:2) Γ ν − (cid:3) = number of eigenvalues of H ν , where the r.h.s. are also equal to − Index( W − m,κ ) and − Index( W ν − ) , respectively. Note that the l.h.s. of the above statement contains four contributions, onefor each function living on the edges of the square. As already mentioned, thecontribution of Γ corresponds to the one of the scattering operator. In addition,the contribution due to Γ and to Γ correspond to corrections to Levinson’stheorem. Explanations on these corrections have been provided in [11] and arequite common in any statement about Levinson’s theorem. In our approach, thesecorrections are automatically taken into account. Remark 5.4.
The wave operators described in the cases I.2), I.3) and I.5) ofProposition 5.1 also belong to E o and the corresponding functions Γ − m,κ or Γ ν − are well defined. However, since these functions vanish at one point on (cid:3) theirwinding numbers are no more well-defined. Accordingly, the corresponding waveoperators are not Fredholm operators, and thus their analytic indexes are also notwell defined. number corresponds to number of times the function t f ( t ) turns around 0 ∈ C when t runson γ . E n for n >
0. Clearly, this algebradoes not contain K R , and thus the previous construction does not apply. In fact,this algebra contains all pseudo-differential operators of order 0 with periodiccoefficients. In such a case the ideal K R has to be replaced by the ideal J n definedby J n := C ∗ (cid:16) a ( D ) b ( X ) | a ∈ C ( R ) , b ∈ C πn ( R ) (cid:17) . Then, the quotient algebra E n / J n can easily be computed and is isomorphic to C πn ( R ) ⊕ C πn ( R ). The quotient morphism q n : E n → C πn ( R ) ⊕ C πn ( R ) is uniquelydetermined by q n (cid:0) a ( D ) b ( X ) (cid:1) = (cid:0) a ( −∞ ) b ( · ) , a (+ ∞ ) b ( · ) (cid:1) . Now, let us consider n ∈ R ∗ and compute the image of W − m,κ by this quotientmap whenever W − m,κ belongs to E | n | , namely in the cases II.1) - II.3) of Proposition5.1. More precisely, for the operator W − in,κ one has q | n | ( W − in,κ ) = (cid:16) i e πn − ς e − πn e inx − ς e + πn e inx , (cid:17) . Note then that since C πn ( R ) can naturally be identified with C ( S ), we definethrough this identification the winding number Wind πn ( f ) of any bounded andboundedly invertible element f ∈ C πn ( R ). Clearly, this is well defined if and onlyif ( in, κ ) is not an exceptional pair.In order to define an analytic index for W − in,κ we recall the construction pro-vided in [5, Sec. 4] about the direct integral decomposition of L ( R ) useful forperiodic systems, the so-called Floquet-Bloch decomposition . More informationcan also be found in [10, Sec.XIII.16]. For simplicity, we provide only the con-struction for n >
0, but the general case can be obtained by replacing n with | n | .For each θ ∈ [0 , n ) we set H θ := L (cid:0) [0 , πn ] , d x (cid:1) endowed with the usual Lebesguemeasure, and also define H n := Z ⊕ [0 , n ) H θ d θ n . Then, if S ( R ) denotes the Schwartz space on R , the map U n : L ( R ) → H n definedfor θ ∈ [0 , n ) and x ∈ [0 , πn ) by[ U n f ]( θ, x ) := X k ∈ Z e − i πn kθ f (cid:16) x + πn k (cid:17) ∀ f ∈ S ( R ) , extends continuously to a unitary operator. The adjoint operator is then given bythe formula [ U ∗ n ϕ ] (cid:0) x + πn k (cid:1) = Z n e i πn kθ ϕ ( θ, x ) d θ n . Moreover, one has U n D U ∗ n = Z ⊕ [0 , n ) D ( θ ) x d θ n , D ( θ ) x is the operator − i dd x on a fiber H θ with boundary condition f ( πn ) =e i πn θ f (0).Thus, for any operator of the form a ( D ) b ( X ) with a ∈ C (cid:0) [ −∞ , ∞ ] (cid:1) and b ∈ C πn ( R ), the operator U n a ( D ) b ( X ) U ∗ n is a decomposable operator with the fibers a (cid:0) D ( θ ) x (cid:1) b ( X ). On suitable bounded decomposable operator Φ = R ⊕ [0 , n ) Φ( θ ) d θ n wealso define the trace Tr n byTr n (Φ) = Z n tr θ (cid:0) Φ( θ ) (cid:1) d θ n where tr θ is the usual trace on H θ .Before stating our main result for the semi-Fredholm operator W − in,κ , let usrecall that W ∓ m,κ denote the transpose operators of W ∓ m,κ , and that the followingrelations have been proved in [3] in the non exceptional case: W ± m,κ W ∓ m,κ = I and W ∓ m,κ W ± m,κ = I R + ( H m,κ )where I R + ( H m,κ ) is a projection related to the continuous spectrum of H m,κ .More precisely, the subspace spanned by this projection is the image throughthe unitary transformation of the complementary to the subspace spanned by theeigenfunctions of the operator H m,κ . This latter subspace for m = in is eitherinfinite dimensional, or 0-dimensional, as already mentioned in Proposition 3.4. Ifwe set I p ( H m,κ ) := 1 − I R + ( H m,κ ) then one has: Theorem 5.5.
Consider n > and κ ∈ C such that ( in, κ ) is not an exceptionalpair. Then, Wind πn (cid:0) S in,κ (cid:1) = − Tr n (cid:0) I p ( H in,κ ) (cid:1) . Let us emphasize that the l.h.s. corresponds to the natural analytic indexIndex n defined in terms of Tr n and evaluated on W − in,κ . A proof for such a state-ment is provided in [5] but is valid only if H in,κ is self-adjoint. This takes placeif and only if | κ | = 1. For completeness we provide below an adaptation of theproof valid in the more general context of the present paper. We shall show in thisproof that the above equality can only take two values: either − H in,κ hasan infinite number of eigenvalues, or 0 when this operator has no eigenvalue. Proof.
In this proof we assume that κ = 0 since in this case the statement istrivially satisfied. From the equalities | ς | = (cid:12)(cid:12)(cid:12) κ Γ( − in )Γ( in ) (cid:12)(cid:12)(cid:12) = e n ( n ln( | κ | ) together with the equality S in,κ ( x ) = i e πn − ς e − πn e inx − ς e πn e inx one easily infers that Wind πn (cid:0) S in,κ (cid:1) = − n ln( | κ | ) ∈ ( − π, π ) while one hasWind πn (cid:0) S in,κ (cid:1) = 0 if n ln( | κ | ) [ − π, π ]. Since the special case n ln( | κ | ) = ± π isan exceptional situation, it is disregarded.20et us now consider an operator of the form a ( D ) b ( X ) with a ∈ C ( R ) and b ∈ C πn ( R ), and the corresponding operator a (cid:0) D ( θ ) x (cid:1) b ( X ). Since the eigenfunctionsof the operator D ( θ ) x are provided by the functions[0 , πn ) ∋ x r nπ e i ( θ +2 nk ) x ∈ C , k ∈ Z we infer that the Schwartz kernel of the operator a (cid:0) D ( θ ) x (cid:1) b ( X ) is given by K a ( D ( θ ) x ) b ( X ) ( x, y ) = nπ X k ∈ Z a ( θ + 2 nk )e i ( θ +2 nk )( x − y ) b ( y ) . Thus, if b ( X ) a (cid:0) D ( θ ) x (cid:1) is tr θ -trace class for a.e. θ ∈ [0 , n ) we obtaintr θ (cid:0) a (cid:0) D ( θ ) x (cid:1) b ( X ) (cid:1) = X k ∈ Z a ( θ + 2 nk ) × nπ Z πn b ( x ) d x and then Tr n (cid:0) a ( D ) b ( X ) (cid:1) = 12 n Z R a ( ξ ) d ξ × nπ Z πn b ( x ) d x. (5.3)Note that these formulas are valid if a has a fast enough decay, which will bethe case in the sequel. In addition, note also that the last term depends only onthe 0-th Fourier coefficient of the function b . Our next aim is thus to show that[ W − in,κ , W + in,κ ] = −I p ( H in,κ ) can be rewritten in the above form.Recall now that Ξ ( D ) is a unitary operator, with Ξ ( D ) ∗ = Ξ ( − D ). Onealso infers from the definition in (4.1) that Ξ in ( D ) is invertible with Ξ in ( D ) − =Ξ in ( − D ) and that Ξ in ( D ) ∗ = Ξ − in ( − D ). Then one getsΞ in ( D ) − Ξ ( D ) (cid:16) W − in,κ W + in,κ − W + in,κ W − in,κ (cid:17) Ξ ( D ) ∗ Ξ in ( D )= (cid:0) I − ςG + n ( D )e inX (cid:1) F in,κ ( X ) (cid:0) ς e inX G − n ( D ) − I (cid:1) − I, where F in,κ ( X ) := − − ς e − πn e inX )(1 − ς e πn e inX )and G ± n ( ξ ) := Ξ ± in ( − ξ )Ξ ∓ in ( ξ ) . From the identity Γ( z + )Γ( − z + ) = π cos( πz ) one then infers that G ± n ( ξ ) = e ± πn e πξ + e ∓ πn e πξ + e ± πn , and by taking into account the identity [3, (6.10)] written in our framework, namelye inX G − n ( D ) + G + n ( D )e inX = 2 cosh( πn )e inX , (cid:0) I − ςG + n ( D )e inX (cid:1) F in,κ ( X ) (cid:0) ς e inX G − n ( D ) − I (cid:1) − I = − F in,κ ( X ) − ς G + n ( D )e inX F in,κ ( X )e inX G − n ( D )+ ςF in,κ ( X )e inX G − n ( D ) + ςG + n ( D )e inX F in,κ ( X ) − I = − F in,κ ( X ) − ς e inX F in,κ ( X )e inX − ς G + n ( D ) (cid:2) e inX F in,κ ( X )e inX , G − n ( D ) (cid:3) + ςF in,κ ( X ) (cid:0) πn )e inX − G + n ( D )e inX (cid:1) + ςG + n ( D )e inX F in,κ ( X ) − I = − F in,κ ( X ) (cid:0) − ς cosh( πn )e inX + ς e inX (cid:1) − I − ς G + n ( D ) (cid:2) e inX F in,κ ( X )e inX , G − n ( D ) (cid:3) − ς (cid:2) F in,κ ( X ) , G + n ( D ) (cid:3) e inX = − ς G + n ( D ) (cid:2) e inX F in,κ ( X )e inX , G − n ( D ) (cid:3)| {z } =: I in,κ − ς (cid:2) F in,κ ( X ) , G + n ( D ) (cid:3) e inX | {z } =: J in,κ For the last step, observe that since the function F in,κ is a smooth πn -periodicfunction, its Fourier series converges uniformly. We can thus write F in,κ ( X ) = P ℓ ∈ Z c ℓ e inℓX . Using the relation e isX g ( D )e − isX = g ( D − s ), which holds for any g ∈ C b ( R ) and s ∈ R , we obtain I in,κ = ς X ℓ ∈ Z c ℓ G + n ( D ) n G − n (cid:0) D − n ( ℓ + 2) (cid:1) − G − n ( D ) o e in ( ℓ +2) X ,J in,κ = ς X ℓ ∈ Z c ℓ n G + n ( D − nℓ ) − G + n ( D ) o e in ( ℓ +1) X . By applying then formula (5.3) one infers that Tr n ( I in,κ ) = 0 since the 0-th Fouriercoefficient of the corresponding function b is obtained for ℓ = −
2, but the firstfactor vanishes precisely when ℓ = −
2. On the other hand one hasTr n ( J in,κ ) = ςc − n Z n X k ∈ Z n G + n (cid:0) θ + 2 n ( k + 1) (cid:1) − G + n (cid:0) θ + 2 nk ) (cid:1)o d θ = ςc − n G + n ( ∞ ) − G + n ( −∞ ) o = ςc − (e πn − e − πn ) . Finally, by collecting the result obtained so far and by using the cyclicity of thetraces one getsTr n (cid:0) W − in,κ W + in,κ − W + in,κ W − in,κ (cid:1) = Tr n (cid:16) Ξ in ( D ) − Ξ ( D ) (cid:16) W − in,κ W + in,κ − W + in,κ W − in,κ (cid:17) Ξ ( D ) ∗ Ξ in ( D ) (cid:17) = − ςc − (e πn − e − πn ) . For the computation of c − it is enough to observe that if n ln( | κ | ) ∈ ( − π, π )22hen F in,κ ( x ) = − − ς e − πn e inx )(1 − ς e πn e inx )= 1 ς e πn e − inx − ς e − πn e inx − ς − e − πn e − inx = 1 ς e πn e − inx ∞ X j =0 (cid:0) ς e − πn e inx (cid:1) j ∞ X k =0 (cid:0) ς − e − πn e − inx (cid:1) k , from which one infers by considering the diagonal sum that c − = ς − e − πn ∞ X j =0 (cid:0) e − πn (cid:1) j = ς − e − πn − e − πn = ς − πn − e − πn . It follows that − Tr n ( I p (cid:0) H in,κ ) (cid:1) = − n ln( | κ | ) ∈ ( − π, π ). On the other hand,if n ln( | κ | ) > π or if n ln( | κ | ) < − π then one gets by a similar argument that thefunction F in,κ has a Fourier series with coefficient c − equal to 0. In such a caseone gets − Tr n ( I p (cid:0) H in,κ ) (cid:1) = 0, as expected. Remark 5.6.
Let us emphasize that the previous theorem is the first topologicalversion of Levinson’s theorem when an infinite number of eigenvalues is involved.Note however that a generalized Levinson’s theorem involving an infinite numberof bound states already appeared in [9, 14], but it corresponds to a relation betweenthe asymptotic behaviors of the spectral shift function and of the eigenvalues count-ing functions. A deeper understanding of the relation between our result and theresults contained in these papers would certainly be valuable.
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