Spectral and scattering theory of one-dimensional coupled photonic crystals
Giuseppe De Nittis, Massimo Moscolari, Serge Richard, Rafael Tiedra de Aldecoa
SSpectral and scattering theory of one-dimensional coupledphotonic crystals
G. De Nittis ∗ , M. Moscolari † , S. Richard ‡ , R. Tiedra de Aldecoa § Facultad de Matem´aticas & Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile,Av. Vicu˜na Mackenna 4860, Santiago, Chile Dipartimento di Matematica, “La Sapienza” Universit`a di RomaPiazzale Aldo Moro 2, 00185 Rome, Italy Graduate school of mathematics, Nagoya University, Chikusa-ku,Nagoya 464-8602, Japan
E-mails: [email protected], [email protected], [email protected],[email protected]
Abstract
We study the spectral and scattering theory of light transmission in a system consisting of twoasymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by ajunction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, wedetermine the nature of the spectrum and prove the existence and completeness of the wave operatorsof the system.
Keywords:
Spectral theory, scattering theory, Maxwell operators, commutator methods.
Contents ∗ Supported by the Chilean Fondecyt Grant 1190204. † Supported by the National Group of Mathematical Physics (GNFM-INdAM) ‡ Supported by the grant
Topological invariants through scattering theory and noncommutative geometry from NagoyaUniversity, and by JSPS Grant-in-Aid for scientific research C no 18K03328, and on leave of absence from Univ. Lyon,Universit´e Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622Villeurbanne cedex, France. § Supported by the Chilean Fondecyt Grant 1170008. a r X i v : . [ m a t h - ph ] A p r Scattering theory 21
In this paper, we study the propagation of an electromagnetic field p (cid:126)E, (cid:126)H q in an infinite one-dimensionalwaveguide. We assume that (i) the waveguide is parallel to the x -axis of the Cartesian coordinate system;(ii) the electric field varies along the y -axis and is constant on the planes perpendicular to the x -axis,i.e., (cid:126)E p x, y , z, t q “ ϕ E p x, t q p y ; (iii) the magnetic field varies along the z -axis and is constant on theplanes perpendicular to the x -axis, i.e., (cid:126)H p x, y , z, t q “ ϕ H p x, t q p z ; (iv) the waveguide is made of isotropicmedium . Under these assumptions, one has ∇ ˆ (cid:126)E “ pB x ϕ E q p z and ∇ ˆ (cid:126)H “ p´B x ϕ H q p y and the dynamicalsourceless Maxwell equations [13] read as ε B t ϕ E “ ´B x ϕ H µ B t ϕ H “ ´B x ϕ E . (1.1)The scalar quantities ε and µ in (1.1) are the electric permittivity and magnetic permeability, respectively.They are strictly positive functions on R describing the interaction of the waveguide with the electromag-netic field. One can also include in the model effects associated to bi-anisotropic media [19]. In our case,this is achieved by modifying the system (1.1) as follows : ε B t ϕ E ` χ B t ϕ H “ ´B x ϕ H µ B t ϕ H ` χ ˚ B t ϕ E “ ´B x ϕ E . (1.2)The (possibly complex-valued) function χ is called bi-anisotropic coupling term . In the sequel, we willrefer to the triple p ε, µ, χ q as the constitutive functions of the waveguide.Let us first discuss the case of periodic waveguides, also known as one-dimensional photonic crystals,consisting in one-dimensional media with dielectric properties which vary periodically in space [14, 21, 40].Mathematically, this translates into the fact that the functions ε, µ and χ in (1.2) are periodic, all with thesame period. This makes (1.2) into a coupled system of differential equations with periodic coefficients,and standard techniques like Bloch-Floquet theory (see e.g. [17]) can be used to study the propagationof solutions (or modes). One of the fundamental properties of periodic waveguides is the presence of afrequency spectrum made of bands and gaps. This implies that not all the modes can propagate alongthe medium, since the propagation of modes associated to frequencies inside a gap is forbidden by the“geometry” of the system. This phenomenon is similar to the one appearing in the theory of periodicSchr¨odinger operators, where one has electronic energy bands instead of frequency bands [24, Sec. XIII.16].The study of the propagation of light in a periodic waveguide can be performed using Bloch-Floquettheory. The situation becomes more complicated when one wants to study the propagation of lightthrough two periodic waveguides of different periods that are connected by a junction. Such a system isschematically represented in Figure 1. The asymptotic behaviour of the system on the left is characterisedby the periodic constitutive functions p ε (cid:96) , µ (cid:96) , χ (cid:96) q , whereas the asymptotic behaviour of the system on theright is characterised by periodic constitutive functions p ε r , µ r , χ r q . Namely, ε Ñ ε (cid:96) when x Ñ ´8 and ε Ñ ε r when x Ñ `8 , and similarly for the other two functions µ and χ (see Assumption 2.2 for a precise The interaction between the electromagnetic field and the dielectric medium is characterised by the electric permittivitytensor ε and the magnetic permeability tensor µ . In an isotropic medium these tensors are multiple of the identity, and thusdetermined by two scalars. In the general theory of bi-anisotropic media, χ is a tensor rather than a scalar. The system of equations (1.2) correspondsto a particular choice of the form of this tensor. For more details on the theory of bi-anisotropic media, we refer the interestedreader to the monograph [19]. (cid:96) onthe left and the other with periodicity of type r on the right. Accordingly, the analysis of the dynamicsof the full system can be performed with the tools of spectral and scattering theories, leading us exactlyto the main goal of this work : the spectral and scattering analysis of one-dimensional coupled photoniccrystals. Figure 1: Two periodic waveguides (one-dimensional photonic crystals) connected by a junctionSince quantum mechanics provides a rich toolbox for the study of problems associated to Schr¨odingerequations, we recast our equations of motion in a Schr¨odinger form to take advantage of these tools, inparticular commutator methods which will be used extensively in this paper. Namely, with the notation w : “ ` ε χχ ˚ µ ˘ ´ for the positive-definite matrix of weights associated to the constitutive functions ε , µ , χ (Maxwell weight for short), we rewrite the system of equations (1.2) in the matrix form i B t ˆ ϕ E ϕ H ˙ “ w ˆ ´ i B x ´ i B x ˙ ˆ ϕ E ϕ H ˙ , (1.3)so that it can be considered as a Schr¨odinger equation for the state p ϕ E , ϕ H q T in the Hilbert spaceL p R , C q . This observation is by no means new. Since the dawn of quantum mechanics, the foundingfathers were well-aware that the Maxwell equations in vacuum are relativistically covariant equations fora massless spin-1 particle [35, pp. 151 & 198]. Moreover, similar Schr¨odinger formulations have alreadybeen employed in the literature to study the quantum scattering theory of electromagnetic waves andother classical waves in homogeneous media [5, 15, 23, 33, 37], and to study the propagation of light inperiodic media [7, 8, 10, 18], among other things. However, to the best of our knowledge, the specificproblem we want to tackle in the present work has never been considered in the literature.The papers [23, 33, 37] deal with the scattering theory of three-dimensional electromagnetic wavesin a homogeneous medium. In that setup, the constitutive tensors ε , µ , χ are asymptotically constant.In contrast, in our one-dimensional setup the constitutive functions ε , µ , χ are only assumed to beasymptotically periodic. This introduces a significant complication and novelty to the model, even thoughit has lower dimension than the three-dimensional models. Also, several works dealing with the scatteringtheory of electromagnetic waves are conducted under the simplifying assumption that χ “ w (Assumption 2.2) and we define the full Hamiltonian M in the Hilbert space H w describing theone-dimensional coupled photonic crystal. In Section 2.2, we define the free Hamiltonian M in the Hilbertspace H associated to M , and we define the operator J : H Ñ H w modelling the junction depictedin Figure 1 (Definition 2.4). The operator M is the direct sum of an Hamiltonian M (cid:96) describing theperiodic waveguide asymptotically on the left and an Hamiltonian M r describing the periodic waveguideasymptotically on the right. In Section 2.3, we use Bloch-Floquet theory to show that the asymptoticHamiltonians M (cid:96) and M r fiber analytically in the sense of G´erard and Nier [11] (Proposition 2.6). As a by-product, we prove that M (cid:96) and M r do not possess flat bands, and thus have purely absolutely continuousspectra (Proposition 2.8). The analytic fibration of M (cid:96) and M r provides also a natural definition for theset T M of thresholds in the spectrum of M (Eqs. (2.5)-(2.6)). In section 3.1, we recall from [1, 29] thenecessary abstract results on commutator methods for self-adjoint operators. In section 3.2, we constructfor each compact interval I Ă R z T M a conjugate operator A ,I for the free Hamiltonian M and use it toprove a limiting absorption principle for M in I (Theorem 3.3 and the discussion that follows). In Section3.3, we use the fact that A ,I is a conjugate operator for M and abstract results on the Mourre theory ina two-Hilbert spaces setting [28] to show that the operator A I : “ JA ,I J ˚ is a conjugate operator for M (Theorem 3.9). In Section 3.4, we use the operator A I to prove a limiting absorption principle for M in I ,which implies in particular that in any compact interval I Ă R z T M the Hamiltonian M has at most finitelymany eigenvalues, each one of finite multiplicity, and no singular continuous spectrum (Theorem 3.15).Using Zhislin sequences (a particular type of Weyl sequences), we also show in Proposition 3.10 that M and M have the same essential spectrum. In Section 4.1, we recall abstract criteria for the existence andthe completeness of wave operators in a two-Hilbert spaces setting. Finally, in Section 4.2, we use theseabstract results in conjunction with the results of the previous sections to prove the existence and thecompleteness of wave operators for the pair t M , M u (Theorem 4.6). We also give an explicit descriptionof the initial sets of the wave operators in terms of the asymptotic velocity operators for the Hamiltonians M (cid:96) and M r (Proposition 4.8 & Theorem 4.10). In this section, we introduce the full Hamiltonian M that we will study. It is a one-dimensional Maxwell-likeoperator describing perturbations of an anisotropic periodic one-dimensional photonic crystal.Throughout the paper, for any Hilbert space H , we write x ¨ , ¨ y H for the scalar product on H , } ¨ } H for the norm on H , B p H q for the set of bounded operators on H , and K p H q for the set of compactoperators on H . We also use the notation B p H , H q (resp. K p H , H q ) for the set of bounded (resp.compact) operators from a Hilbert space H to a Hilbert space H . Definition 2.1 (One-dimensional Maxwell-like operator) . Let ă c ă c ă 8 and take a Hermitianmatrix-valued function w P L ` R , B p C q ˘ such that c ď w p x q ď c for a.e. x P R . Let P be the omentum operator in L p R q , that is, P f : “ ´ if for each f P H p R q , with H p R q the first Sobolevspace on R . Let Dϕ : “ ˆ PP ˙ ϕ, ϕ P D p D q : “ H p R , C q . Then, the Maxwell-like operator M in L p R , C q is defined as Mϕ : “ w Dϕ, ϕ P D p M q : “ D p D q . The Maxwell weight w that we consider converges at ˘8 to periodic functions in the following sense : Assumption 2.2 (Maxwell weight) . There exist ε ą and hermitian matrix-valued functions w (cid:96) , w r P L ` R , B p C q ˘ periodic of periods p (cid:96) , p r ą such that ›› w p x q ´ w (cid:96) p x q ›› B p C q ď Const . x x y ´ ´ ε , a.e. x ă , ›› w p x q ´ w r p x q ›› B p C q ď Const . x x y ´ ´ ε , a.e. x ą , (2.1) where the indexes (cid:96) and r stand for “left” and “right”, and x x y : “ p ` | x | q { . Lemma 2.3.
Let Assumption 2.2 be satisfied.(a) One has for a.e. x P R the inequalities c ď w (cid:96) p x q ď c and c ď w r p x q ď c , with c , c introduced in Definition 2.1.(b) The sesquilinear form x¨ , ¨y H w : L p R , C q ˆ L p R , C q Ñ C , p ψ, ϕ q ÞÑ @ ψ, w ´ ϕ D L p R , C q , defines a new scalar product on L p R , C q , and we denote by H w the space L p R , C q equippedwith x ¨ , ¨ y H w . Moreover, the norm of L p R , C q and H w are equivalent, and the claim remains trueif one replace w with w (cid:96) or w r .(c) The operator M with domain D p M q : “ H p R , C q is self-adjoint in H w .Proof. Point (a) is a direct consequence of the assumptions on w , w (cid:96) , w r . Point (b) follows from thebounds c ď w p x q , w (cid:96) p x q , w r p x q ď c valid for a.e. x P R . Point (c) can be proved as in [9, Prop. 6.2]. We now define the free Hamiltonian associated to the operator M . Due to the anisotropy of the Maxwellweight w at ˘8 , it is convenient to define left and right asymptotic operators M (cid:96) : “ w (cid:96) D and M r : “ w r D, with w (cid:96) and w r as in Assumption 2.2. Lemma 2.3(c) implies that the operators M (cid:96) and M r are self-adjointin the Hilbert spaces H w (cid:96) and H w r , with the same domain D p M (cid:96) q “ D p M r q “ D p M q . Then, we define thefree Hamiltonian as the direct sum operator M : “ M (cid:96) ‘ M r in the Hilbert space H : “ H w (cid:96) ‘ H w r . Since the free Hamiltonian acts in the Hilbert space H and thefull Hamiltonian acts in the Hilbert space H w , we need to introduce an identification operator betweenthe spaces H and H w : 5 efinition 2.4 (Junction operator) . Let j (cid:96) , j r P C p R , r , sq be such that j (cid:96) p x q : “ if x ď ´ if x ě ´ { and j r p x q : “ if x ď { if x ě .Then, J : H Ñ H w is the bounded operator defined by J p ϕ (cid:96) , ϕ r q : “ j (cid:96) ϕ (cid:96) ` j r ϕ r , with adjoint J ˚ : H w Ñ H given by J ˚ ϕ “ ` w (cid:96) w ´ j (cid:96) ϕ, w r w ´ j r ϕ ˘ . Remark 2.5.
We call J the junction operator because it models mathematically the junction depictedin Figure 1. Indeed, the Hamiltonian M only describes the free dynamics of the system in the bulkasymptotically on the left and in the bulk asymptotically on the right. Since M is the direct sum of theoperators M (cid:96) and M r , the interface effects between the left and the right parts of the system are notdescribed by M in any way. The role of the operator J is thus to map the free bulk states of the systembelonging to the direct sum Hilbert space H onto a joined state belonging to the physical Hilbert space H w , where acts the full Hamiltonian M describing the interface effects.Given a state ϕ P H w , the square norm E p ϕ q : “ } ϕ } H w can be interpreted as the total energy ofthe electromagnetic field ϕ ” p ϕ E , ϕ H q T . A direct computation shows that the total energy of a state J p ϕ (cid:96) , ϕ r q obtained by joining bulk states ϕ (cid:96) P H (cid:96) and ϕ r P H r satisfies E ` J p ϕ (cid:96) , ϕ r q ˘ “ E (cid:96) p ϕ (cid:96) q ` E r p ϕ r q ` E interface p ϕ (cid:96) , ϕ r q with E (cid:96) p ϕ (cid:96) q : “ } j (cid:96) ϕ (cid:96) } H w(cid:96) and E r p ϕ r q : “ } j r ϕ r } H w r the total energies of the field j (cid:96) ϕ (cid:96) on the left and thefield j r ϕ r on the right, and with E interface p ϕ (cid:96) , ϕ r q : “ @ ϕ (cid:96) , j (cid:96) ` w ´ ´ w ´ (cid:96) ˘ j (cid:96) ϕ (cid:96) D L p R , C q ` @ ϕ r , j r ` w ´ ´ w ´ ˘ j r ϕ r D L p R , C q the energy associated with the left and right external interfaces of the junction. In particular, one noticesthat there is no contribution to the energy associated to the central region p´ { , { q of the junction.This physical observation shows as a by-product that the operator J is neither invertible, nor isometric. In this section, we introduce a Bloch-Floquet (or Bloch-Floquet-Zak or Floquet-Gelfand) transform totake advantage of the periodicity of the operators M (cid:96) and M r . For brevity, we use the symbol ‹ to denoteeither the index “ (cid:96) ” or the index “r”.Let Γ ‹ : “ (cid:32) np ‹ | n P Z ( Ă R be the one-dimensional lattice of period p ‹ with fundamental cell Y ‹ : “ r´ p ‹ { , p ‹ { s , and letΓ ˚‹ : “ (cid:32) πn { p ‹ | n P Z ( Ă R be the reciprocal lattice of Γ ‹ with fundamental cell Y ˚‹ : “ r´ π { p ‹ , π { p ‹ s . For each t P R , we define thetranslation operator T t : L p R , C q Ñ L p R , C q , ϕ ÞÑ ϕ p ¨ ´ t q . Using this operator, we can define the Bloch-Floquet transform of a C -valued Schwartz function ϕ P S p R , C q as p U ‹ ϕ qp k, θ q : “ ÿ n P Z e ´ ik p θ ´ np ‹ q ` T np ‹ ϕ ˘ p θ q , k, θ P R . U ‹ ϕ is p ‹ -periodic in the variable θ , p U ‹ ϕ qp k, θ ` γ q “ p U ‹ ϕ qp k, θ q , γ P Γ ‹ , and 2 π { p ‹ -pseudo-periodic in the variable k , p U ‹ ϕ qp k ` γ ˚ , θ q “ e ´ iθγ ˚ p U ‹ ϕ qp k, θ q , γ ˚ P Γ ˚‹ . Now, let h ‹ be the Hilbert space obtained by equipping the set (cid:32) ϕ P L p R , C q | T γ ϕ “ ϕ for all γ P Γ ‹ ( with the scalar product x ϕ, ψ y h ‹ : “ ż Y ‹ d θ @ ϕ p θ q , w ‹ p θ q ´ ψ p θ q D C . Since h ‹ and L p Y ‹ , C q are isomorphic, we shall use both representations. Next, let τ : Γ ˚‹ Ñ B p h ‹ q bethe unitary representation of the dual lattice Γ ˚‹ on h ‹ given by ` τ p γ ˚ q ϕ ˘ p θ q : “ e iθγ ˚ ϕ p θ q , γ ˚ P Γ ˚‹ , ϕ P h ‹ , a.e. θ P R ,and let H τ, ‹ be the Hilbert space obtained by equipping the set (cid:32) u P L p R , h ‹ q | u p ¨ ´ γ ˚ q “ τ p γ ˚ q u for all γ ˚ P Γ ˚‹ ( with the scalar product x u, v y H τ, ‹ : “ | Y ˚‹ | ż Y ˚‹ d k @ u p k q , v p k q D h ‹ . There is a natural isomorphism from H τ, ‹ to L p Y ˚‹ , h ‹ q given by the restriction from R to Y ˚‹ , and withinverse given by τ -equivariant continuation. However, using H τ, ‹ has various advantages and we shallstick to it in the sequel. Direct calculations show that the Bloch-Floquet transform extends to a unitaryoperator U ‹ : H w ‹ Ñ H τ, ‹ with inverse ` U ´ ‹ u ˘ p x q “ | Y ˚‹ | ż Y ˚‹ d k e ikx ` u p k q ˘ p x q , u P H τ, ‹ , a.e. x P R .Furthermore, since M ‹ commutes with the translation operators T γ ( γ P Γ ‹ ), the operator M ‹ is decom-posable in the Bloch-Floquet representation. Namely, we have x M ‹ : “ U ‹ M ‹ U ´ ‹ “ (cid:32) x M ‹ p k q ( k P R with x M ‹ p k ´ γ ˚ q “ τ p γ ˚ q x M ‹ p k q τ p γ ˚ q ˚ , k P R , γ ˚ P Γ ˚‹ , (2.2)and x M ‹ p k q “ w ‹ p D p k q and p D p k q u p k q “ ˆ ´ i B θ ` k ´ i B θ ` k ˙ u p k q , k P Y ˚‹ , u P U ‹ D p M ‹ q . Here, the domain U ‹ D p M ‹ q of U ‹ M ‹ U ´ ‹ satisfies U ‹ D p M ‹ q “ U ‹ H p R , C q “ (cid:32) u P L p R , h ‹ q | u p ¨ ´ γ ˚ q “ τ p γ ˚ q u for all γ ˚ P Γ ˚‹ ( with h ‹ : “ (cid:32) ϕ P H p R , C q | T γ ϕ “ ϕ for all γ P Γ ‹ ( . In the next proposition, we prove that the operator x M ‹ is analytically fibered in the sense of [11,Def. 2.2]. For this, we need to introduce the Bloch varietyΣ ‹ : “ (cid:32) p k, λ q P Y ˚‹ ˆ R | λ P σ ` x M ‹ p k q ˘( . (2.3)7 roposition 2.6 (Fibering of the asymptotic Hamiltonians) . Let x M ‹ p ω q ϕ : “ ˆ w ‹ p D p q ` w ‹ ˆ ωω ˙˙ ϕ, ω P C , ϕ P h ‹ . (a) The set O ‹ : “ (cid:32) p ω, z q P C ˆ C | z P ρ ` x M ‹ p ω q ˘( . is open in C ˆ C and the map O ‹ Q p ω, z q ÞÑ ` x M ‹ p ω q ´ z ˘ ´ P B p h ‹ q is analytic in the variables ω and z .(b) For each ω P C , the operator x M ‹ p ω q has purely discrete spectrum.(c) If Σ ‹ is equipped with the topology induced by Y ˚‹ ˆ C , then the projection π R : Σ ‹ Ñ R given by π R p k, λ q : “ λ is proper.In particular, the operator x M ‹ is analytically fibered in the sense of [11, Def. 2.2].Proof. (a) The operator w ‹ p D p q is self-adjoint on h ‹ Ă h ‹ , and for each ω P C we have that w ‹ ` ωω ˘ P B p h ‹ q . Hence, for each ω P C the operator x M ‹ p ω q is closed in h ‹ and has domain h ‹ , and for each x P R the operator x M ‹ p x q is self-adjoint on h ‹ . In particular, we infer by functional calculus thatlim | t |Ñ8 ›››` x M ‹ p x q ´ it ˘ ´ ››› B p h ‹ q ď lim | t |Ñ8 | t | “ p t P R q . Therefore, for each y P R the setΩ y : “ " it P i R | ´››` x M ‹ p x q ´ it ˘ ´ ›› B p h ‹ q ¯ ´ ą | y | } w ‹ } B p h ‹ q * is non-empty, and then the argument in [16, Rem. IV.3.2] guarantees that Ω y is contained in the resolventset of x M ‹ p x ` iy q . Thus, for each ω ” x ` iy P C the operator x M ‹ p ω q is closed in h ‹ , has domain h ‹ ,and non-empty resolvent set, and for each ϕ P h ‹ the map C Q ω ÞÑ x M ‹ p ω q ϕ P h ‹ is linear and thereforeanalytic. So, the collection (cid:32) x M ‹ p ω q ( ω P C is an analytic family of type (A) [24, p. 16], and thus also ananalytic family in the sense of Kato [24, p. 14]. The claim is then a consequence of [24, Thm. XII.7].(b) Since (cid:32) x M ‹ p ω q ( ω P C is an analytic family of type (A), the operators x M ‹ p ω q have compact resolvent(and thus purely discrete spectrum) either for all ω P C or for no ω P C [16, Thm. III.6.26 & VII.2.4].Therefore, to prove the claim, it is sufficient to show that x M ‹ p q has compact resolvent. Now, we have x M ‹ p q “ w ‹ p D p q “ w ‹ ˆ ´ i B θ ´ i B θ ˙ , where ´ i B θ is a first order differential operator in L p Y ‹ q with periodic boundary conditions, and thus withpurely discrete spectrum that accumulates at ˘8 . In consequence, each entry of the matrix operator ` p D p q ` i ˘ ´ “ ˜ ´ i ` ` p i B θ q ˘ ´ ´ i B θ ` ` p i B θ q ˘ ´ ´ i B θ ` ` p i B θ q ˘ ´ ´ i ` ` p i B θ q ˘ ´ ¸ is compact in L p Y ‹ q , so that ` p D p q ` i ˘ ´ is compact in L p Y ‹ , C q . Since Lemma 2.3(a) implies thatthe norms on L p Y ‹ , C q and h ‹ are equivalent, we infer that ` p D p q ` i ˘ ´ is also compact in h ‹ . Finally,since ` x M ‹ p q ` i ˘ ´ “ ` x M ‹ p q ` iw ‹ ˘ ´ ` ` x M ‹ p q ` i ˘ ´ ´ ` x M ‹ p q ` iw ‹ ˘ ´ “ ` p D p q ` i ˘ ´ w ´ ‹ ´ i ` x M ‹ p q ` i ˘ ´ p ´ w ‹ q ` p D p q ` i ˘ ´ w ´ ‹ , (2.4)8ith w ´ ‹ and p ´ w ‹ q bounded in h ‹ , we obtain that ` x M ‹ p q ` i ˘ ´ is compact in h ‹ .(c) Let Y ˚‹ ˆ C be endowed with the topology induced by C ˆ C . Point (a) implies that the setΣ c ‹ : “ (cid:32) p k, z q P Y ˚‹ ˆ C | z P ρ ` x M ‹ p k q ˘( is open in Y ˚‹ ˆ C . Therefore, the set Σ ‹ is closed in Y ˚‹ ˆ C and the inclusion ι : Σ ‹ Ñ Y ˚‹ ˆ C is a closedmap. Since the projection π C : Y ˚‹ ˆ C Ñ C given by π C p k, z q : “ z is also a closed map (because Y ˚‹ iscompact, see [20, Ex. 7, p. 171]) and π R “ π C ˝ ι , we infer that π R is a closed map. Moreover, π R iscontinuous because it is the restriction to the subset Σ ‹ of the continuous projection π C : Y ˚‹ ˆ C Ñ C .In consequence, in order to prove that π R is proper it is sufficient to show that π ´ R pt λ uq is compact inΣ ‹ for each λ P π R p Σ ‹ q . But since π ´ R pt λ uq “ ` ι ´ ˝ π ´ C ˘ pt λ uq “ ι ´ p Y ˚‹ ˆ t λ uq “ p Y ˚‹ ˆ t λ uq X Σ ‹ , this follows from compactness of Y ˚‹ and the closedness of Σ ‹ in Y ˚‹ ˆ C .Proposition 2.6 can be combined with the theorem of Rellich [16, Thm. VII.3.9] which, adapted toour notations, states : Theorem 2.7 (Rellich) . Let Ω Ă C be a neighborhood of an interval I Ă R and let t T p ω qu ω P Ω be a self-adjoint analytic family of type (A), with each T p ω q having compact resolvent. Then, there is a sequenceof scalar-valued functions λ n and a sequence of vector-valued functions u n , all analytic on I , such thatfor ω P I the λ n p ω q are the repeated eigenvalues of T p ω q and the u n p ω q form a complete orthonormalfamily of the associated eigenvectors of T p ω q . By applying this theorem to the family (cid:32) x M ‹ p ω q ( ω P C , we infer the existence of analytic eigenvaluefunctions λ ‹ ,n : Y ˚‹ Ñ R and analytic orthonormal eigenvector functions u ‹ ,n : Y ˚‹ Ñ h ‹ . We call bandthe graph (cid:32)` k, λ ‹ ,n p k q ˘ | k P Y ˚‹ ( of the eigenvalue function λ ‹ ,n , so that the Bloch variety Σ ‹ coincideswith the countable union of the bands (see (2.3)). Since the derivative λ ,n of λ ‹ ,n exists and is analytic,it is natural to define the set of thresholds of the operator M ‹ as T ‹ : “ ď n P N (cid:32) λ P R | D k P Y ˚‹ such that λ “ λ ‹ ,n p k q and λ ,n p k q “ ( , (2.5)and the set of thresholds of both M (cid:96) and M r as T M : “ T (cid:96) Y T r . (2.6)Proposition 2.6(b), together with the analyticity of the functions λ ‹ ,n , implies that the set T ‹ is discrete,with only possible accumulation point at infinity. Furthermore, [24, Thm. XIII.85(e)] implies that thepossible eigenvalues of M ‹ are contained in T ‹ . However, these eigenvalues should be generated by locally(hence globally) flat bands, and one can show their absence by adapting Thomas’ argument [34, Sec. II]to our setup : Proposition 2.8 (Spectrum of the asymptotic Hamiltonians) . The spectrum of M ‹ is purely absolutelycontinuous. In particular, σ p M ‹ q “ σ ac p M ‹ q “ σ ess p M ‹ q , with σ ac p M ‹ q the absolutely continuous spectrum of M ‹ and σ ess p M ‹ q the essential spectrum of M ‹ .Proof. In view of [24, Thm. XIII.86], the claim follows once we prove the absence of flat bands for M ‹ .For this purpose, we use the version of the Thomas’ argument as presented in [31, Sec. 1.3]. Accordingly,we first need to show that, for ω “ iρ with ρ P R large enough, the operator x M ‹ p iρ q is invertible andsatisfies lim | ρ |Ñ8 ›› x M ‹ p iρ q ´ ›› B p h ‹ q “ . (2.7)9et us start with the operator p D p iρ q “ ˆ ´ i B θ ` iρ ´ i B θ ` iρ ˙ acting on h ‹ Ă h ‹ . Since the family of functions t e ˘ n u n P Z given by e ` n p θ q : “ ? p ‹ e πinθ { p ‹ ˆ ˙ , e ´ n p θ q : “ ? p ‹ e πinθ { p ‹ ˆ ˙ , θ P Y ‹ , is an orthonormal basis of L p Y ‹ , C q , and since h ‹ and L p Y ‹ , C q have equivalent norms, the family t e ˘ n u n P Z , with extended variable θ P R , is also a (non-orthogonal) basis for h ‹ , and thus any ϕ P h ‹ canbe expanded in h ‹ as ϕ “ ÿ n P Z ` p ϕ ` n e ` n ` p ϕ ´ n e ´ n ˘ with p ϕ ˘ n : “ @ ϕ, e ˘ n D L p Y ‹ , C q . It follows that ›› p D p˘ iρ q ϕ ›› h ‹ “ ››››› ÿ n P Z ´ πnp ‹ ˘ iρ ¯ ` p ϕ ` n e ´ n ` p ϕ ´ n e ` n ˘››››› h ‹ ě Const . ›››››ÿ n P Z ´ πnp ‹ ˘ iρ ¯ ` p ϕ ` n e ´ n ` p ϕ ´ n e ` n ˘››››› p Y ‹ , C q “ Const . ÿ n P Z ˇˇˇ πnp ‹ ˘ iρ ˇˇˇ ` | p ϕ ` n | ` | p ϕ ´ n | ˘ ě Const . | ρ | } ϕ } p Y ‹ , C q ě Const . | ρ | } ϕ } h ‹ . Thus, the operators p D p˘ iρ q are injective with closed range and satisfy in h ‹ the relations ` Ran p D p˘ iρ q ˘ K “ Ker ` p D p˘ iρ q ˚ ˘ “ Ker ` w ‹ p D p¯ iρ q w ´ ‹ ˘ “ . In consequence Ran p D p˘ iρ q “ h ‹ , and the operators p D p˘ iρ q are invertible with ›› p D p˘ iρ q ´ ›› B p h ‹ q ď Const . | ρ | ´ . It follows that x M ‹ p iρ q is invertible too, with ›› x M ‹ p iρ q ´ ›› B p h ‹ q “ ›› p D p iρ q ´ w ´ ‹ ›› B p h ‹ q ď Const . | ρ | ´ , which implies (2.7).Now, let us assume by contradiction that there exists n P N such that λ ‹ ,n p k q is equal to a constant c P R for all k P Y ˚‹ . Then, using the analyticity properties of x M ‹ (Proposition 2.6) in conjunction withthe analytic Fredholm alternative, one infers that c is an eigenvalue of x M ‹ p ω q for all ω P C . Letting u p ω q be the corresponding eigenfunction for x M ‹ p ω q , one obtains that x M ‹ p ω q u p ω q “ cu p ω q for all ω P C .Choosing ω “ iρ with ρ P R and using the fact that x M ‹ p iρ q is invertible, one thus obtains that u p iρ q “ c x M ‹ p iρ q ´ u p iρ q with } u p iρ q} h ‹ “ , which contradicts (2.7). Remark 2.9.
The absence of flat bands for the 3-dimensional Maxwell operator has been discussed in [31,Sec. 5]. However, the results of [31] do not cover the result of Proposition 2.8 since the weights consideredin [31] are block-diagonal and smooth while in Proposition 2.8 the weights are L positive-definite ˆ matrices. Neither diagonality, nor smoothness is assumed. Mourre theory and spectral results
In this section, we recall some definitions appearing in Mourre theory and provide a precise meaning tothe commutators mentioned in the introduction. We refer to [1, 29] for more information and details.Let A be a self-adjoint operator in a Hilbert space H with domain D p A q , and let T P B p H q . For any k P N , we say that T belongs to C k p A q , with notation T P C k p A q , if the map R Q t ÞÑ e ´ itA T e itA P B p H q (3.1)is strongly of class C k . In the case k “
1, one has T P C p A q if and only if the quadratic form D p A q Q ϕ ÞÑ x ϕ, T A ϕ y H ´ x A ϕ, T ϕ y H P C is continuous for the norm topology induced by H on D p A q . We denote by r T, A s the bounded operatorassociated with the continuous extension of this form, or equivalently ´ i times the strong derivative ofthe function (3.1) at t “ H is a self-adjoint operator in H with domain D p H q and spectrum σ p H q , we say that H is of class C k p A q if p H ´ z q ´ P C k p A q for some z P C z σ p H q . In particular, H is of class C p A q if and only if thequadratic form D p A q Q ϕ ÞÑ @ ϕ, p H ´ z q ´ A ϕ D H ´ @ A ϕ, p H ´ z q ´ ϕ D H P C extends continuously to a bounded form defined by the operator “ p H ´ z q ´ , A ‰ P B p H q . In such a case,the set D p H q X D p A q is a core for H and the quadratic form D p H q X D p A q Q ϕ ÞÑ x
Hϕ, A ϕ y H ´ x A ϕ, Hϕ y H P C is continuous in the natural topology of D p H q ( i.e. the topology of the graph-norm) [1, Thm. 6.2.10(b)].This form then extends uniquely to a continuous quadratic form on D p H q which can be identified with acontinuous operator r H, A s from D p H q to the adjoint space D p H q ˚ . In addition, one has the identity “ p H ´ z q ´ , A ‰ “ ´p H ´ z q ´ r H, A sp H ´ z q ´ , (3.2)and the following result is verified [1, Thm. 6.2.15] : If H is of class C k p A q for some k P N and η P S p R q is a Schwartz function, then η p H q P C k p A q .A regularity condition slightly stronger than being of class C p A q is defined as follows : H is of class C ` ε p A q for some ε P p , q if H is of class C p A q and if for some z P C z σ p H q ›› e ´ itA “ p H ´ z q ´ , A ‰ e itA ´ “ p H ´ z q ´ , A ‰›› B p H q ď Const . t ε for all t P p , q .The condition C p A q is stronger than C ` ε p A q , which in turn is stronger than C p A q .We now recall the definition of two useful functions introduced in [1, Sec. 7.2]. For this, we needthe following conventions : if E H p ¨ q denotes the spectral projection-valued measure of H , then we set E H p λ ; ε q : “ E H ` p λ ´ ε, λ ` ε q ˘ for any λ P R and ε ą
0, and if
S, T P B p H q , then we write S « T if S ´ T is compact, and S À T if there exists a compact operator K such that S ď T ` K . With theseconventions, we define for H of class C p A q the function (cid:37) AH : R Ñ p´8 , by (cid:37) AH p λ q : “ sup (cid:32) a P R | D ε ą a E H p λ ; ε q ď E H p λ ; ε qr iH, A s E H p λ ; ε q ( , and we define the function r (cid:37) AH : R Ñ p´8 , by r (cid:37) AH p λ q : “ sup (cid:32) a P R | D ε ą a E H p λ ; ε q À E H p λ ; ε qr iH, A s E H p λ ; ε q ( . r (cid:37) AH is often useful : r (cid:37) AH p λ q “ sup (cid:32) a P R | D η P C c p R , R q such that η p λ q ‰ a η p H q À η p H qr iH, A s η p H q ( . (3.3)One says that A is conjugate to H at a point λ P R if r (cid:37) AH p λ q ą
0, and that A is strictly conjugate to H at λ if (cid:37) AH p λ q ą
0. It is shown in [1, Prop. 7.2.6] that the function r (cid:37) AH : R Ñ p´8 , is lower semicontinuous,that r (cid:37) AH ě (cid:37) AH , and that r (cid:37) AH p λ q ă 8 if and only if λ P σ ess p H q . In particular, the set of points where A isconjugate to H , r µ A p H q : “ (cid:32) λ P R | r (cid:37) AH p λ q ą ( , is open in R .The main consequences of the existence of a conjugate operator A for H are given in the theorembelow, which is a particular case of [29, Thm. 0.1 & 0.2]. For its statement, we use the notation σ p p H q forthe point spectrum of H , and we recall that if G is an auxiliary Hilbert space, then an operator T P B p H , G q is locally H -smooth on an open set I Ă R if for each compact set I Ă I there exists c I ě ż R d t ›› T e ´ itH E H p I q ϕ ›› G ď c I } ϕ } H for each ϕ P H , (3.4)and T is (globally) H -smooth if (3.4) is satisfied with E H p I q replaced by the identity 1. Theorem 3.1 (Spectrum of H ) . Let
H, A be a self-adjoint operators in a Hilbert space H , let G be anauxiliary Hilbert space, assume that H is of class C ` ε p A q for some ε P p , q , and suppose there exist anopen set I Ă R , a number a ą and an operator K P K p H q such that E H p I q r iH, A s E H p I q ě a E H p I q ` K. (3.5) Then,(a) each operator T P B p H , G q which extends continuously to an element of B ` D px A y s q ˚ , G ˘ for some s ą { is locally H -smooth on I z σ p p H q ,(b) H has at most finitely many eigenvalues in I , each one of finite multiplicity, and H has no singularcontinuous spectrum in I . With the definitions of Section 2.3 at hand, we can construct a conjugate operator for the operator x M ‹ .Our construction follows from the one given in [11, Sec. 3], but it is simpler because our base manifold Y ˚‹ is one-dimensional. Indeed, thanks to Theorem 2.7, it is sufficient to construct the conjugate operatorband by band.So, for each n P N , let p Π ‹ ,n : “ (cid:32)p Π ‹ ,n p k q ( k P R and p λ ,n : “ (cid:32)p λ ,n p k q ( k P R be the bounded decomposableself-adjoint operators in H τ, ‹ defined by τ -equivariant continuation as in (2.2) and by the relations p Π ‹ ,n p k q ϕ : “ @ u ‹ ,n p k q , ϕ D h ‹ u ‹ ,n p k q and p λ ,n p k q ϕ : “ λ ,n p k q ϕ, k P Y ˚‹ , ϕ P h ‹ . Set also Π ‹ ,n : “ U ´ ‹ p Π ‹ ,n U ‹ and x Q ‹ : “ U ‹ Q ‹ U ´ ‹ , with Q ‹ the operator of multiplication by the variablein H w ‹ p Q ‹ ϕ qp x q : “ x ϕ p x q , ϕ P D p Q ‹ q : “ (cid:32) ϕ P H w ‹ | } Q ‹ ϕ } H w ‹ ă 8 ( . Remark 3.2.
Since Q ‹ commutes with w ´ ‹ , the operator Q ‹ is self-adjoint in H w ‹ and essentially self-adjoint on S p R , C q Ă H w ‹ . The definition and the domain of Q ‹ are independent of the specific weight w ´ ‹ appearing in the scalar product of H w ‹ . The insistence on the label ‹ “ (cid:96), r is only motivated by anotational need that will result helpful in the next sections. I Ă R z T ‹ , we define the finite set N p I q : “ (cid:32) n P N | λ ´ ‹ ,n p I q ‰ ∅ ( . Finally,we set D ‹ : “ U ‹ S p R , C q Ă (cid:32) u P C p R , h ‹ q | u p ¨ ´ γ ˚ q “ τ p γ ˚ q u for all γ ˚ P Γ ˚‹ ( . (3.6)Then, we can define the symmetric operator p A ‹ ,I in H τ, ‹ by p A ‹ ,I u : “ ÿ n P N p I q p Π ‹ ,n `p λ ,n x Q ‹ ` x Q ‹ p λ ,n ˘p Π ‹ ,n u, u P D ‹ . (3.7) Theorem 3.3 (Mourre estimate for x M ‹ ) . Let I Ă R z T ‹ be a compact interval. Then,(a) the operator p A ‹ ,I is essentially self-adjoint on D ‹ and on any other core for x Q ‹ , with closure denotedby the same symbol,(b) the operator x M ‹ is of class C p p A ‹ ,I q ,(c) there exists c I ą such that (cid:37) p A ‹ ,I y M ‹ ě c I .Proof. (a) The claim is a consequence of Nelson’s criterion of self-adjointness [22, Thm. X.37] applied tothe triple p p A ‹ ,I , N ‹ , D ‹ q , where N ‹ : “ x Q ‹ ` x Q ‹ : “ U ‹ Q ‹ U ´ ‹ . Indeed, the operator N ‹ is essentiallyself-adjoint on D ‹ “ U ‹ S p R , C q since Q ‹ is essentially self-adjoint on S p R , C q . In addition, since p A ‹ ,I is composed of the bounded operators p Π ‹ ,n and p λ ,n which are analytic in the variable k P Y ˚‹ and x Q ‹ acts as i B k in H τ, ‹ , a direct computation gives ›› p A ‹ ,I u ›› H τ, ‹ ď Const . } N ‹ u } H τ, ‹ , u P D ‹ . Similarly, a direct computation using the boundedness and the analyticity of p Π ‹ ,n and p λ ,n implies that ˇˇˇ@ p A ‹ ,I u, N ‹ u D H τ, ‹ ´ @ N ‹ u, p A ‹ ,I u D H τ, ‹ ˇˇˇ ď Const . x N ‹ u, u y H τ, ‹ , u P D ‹ . In both inequalities, we used the fact that D p x Q ‹ q Ă D p x Q ‹ q . As a consequence, p A ‹ ,I is essentially self-adjoint on D ‹ and on any other core for N ‹ .(b) The set E ‹ : “ (cid:32) u P C p R , h ‹ q | u p ¨ ´ γ ˚ q “ τ p γ ˚ q u for all γ ˚ P Γ ˚‹ ( Ą D ‹ is a core for N ‹ . So, it follows from point (a) that p A ‹ ,I is essentially self-adjoint on E ‹ . Moreover, since x M ‹ p k q is analytic in k P R and satisfies the covariance relation (2.2), we obtain that ` x M ‹ ´ z ˘ E ‹ Ă E ‹ forany z P C z σ p x M ‹ q . Since the same argument applies to the resolvent, we obtain that ` x M ‹ ´ z ˘ ´ E ‹ “ E ‹ .Therefore, we have the inclusion ` x M ‹ ´ z ˘ ´ u P D p p A ‹ ,I q for each u P E ‹ , and a calculation using (3.2)gives @ u, “ i ` x M ‹ ´ z ˘ ´ , p A ‹ ,I ‰ u D H τ, ‹ “ @ u, ´ ` x M ‹ ´ z ˘ ´ “ i x M ‹ , p A ‹ ,I ‰` x M ‹ ´ z ˘ ´ u D H τ, ‹ “ B u, ´ ` x M ‹ ´ z ˘ ´ ÿ n P N p I q p Π ‹ ,n ˇˇp λ ,n ˇˇ p Π ‹ ,n ` x M ‹ ´ z ˘ ´ u F H τ, ‹ . Since ř n P N p I q p Π ‹ ,n ˇˇp λ ,n ˇˇ p Π ‹ ,n P B p H τ, ‹ q , it follows that x M ‹ is of class C p p A ‹ ,I q with “ i x M ‹ , p A ‹ ,I ‰ “ ÿ n P N p I q p Π ‹ ,n ˇˇp λ ,n ˇˇ p Π ‹ ,n . (3.8)13inally, since p Π ‹ ,n P C p p A ‹ ,I q and p λ ,n P C p p A ‹ ,I q for each n P N , we infer from (3.8) and [1, Prop. 5.1.5]that x M ‹ is of class C p p A ‹ ,I q .(c) Using point (b) and the definition of the operators p Π ‹ ,n , we obtain for all η P C c p I, R q and k P Y ˚‹ that η ` x M ‹ p k q ˘“ i x M ‹ , p A ‹ ,I ‰ p k q η ` x M ‹ p k q ˘ “ η ` x M ‹ p k q ˘˜ ÿ n P N p I q p Π ‹ ,n p k q ˇˇp λ ,n p k q ˇˇ p Π ‹ ,n p k q ¸ η ` x M ‹ p k q ˘ ě c I η ` x M ‹ p k q ˘˜ ÿ n P N p I q p Π ‹ ,n p k q ¸ η ` x M ‹ p k q ˘ “ c I η ` x M ‹ p k q ˘ with c I : “ min n P N p I q min t k P Y ˚‹ | λ ‹ ,n p k qP I u ˇˇ λ n p k q ˇˇ . Thus, by using the definition of the scalar product in H τ, ‹ ,we infer that η ` x M ‹ ˘“ i x M ‹ , p A ‹ ,I ‰ η ` x M ‹ ˘ ě c I η ` x M ‹ ˘ , which, together with the definition (3.3), implies the claim.Since the operator p A ‹ ,I is essentially self-adjoint on D ‹ “ U ‹ S p R , C q and on any other core for x Q ‹ , it follows by Theorem 3.3(a) that the inverse Bloch-Floquet transform of p A ‹ ,I , A ‹ ,I : “ U ´ ‹ p A ‹ ,I U ‹ , is essentially self-adjoint on S p R , C q and on any other core for Q ‹ . Therefore, the results (b) and (c) ofTheorem 3.3 can be restated as follows : the operator M ‹ is of class C p A ‹ ,I q and there exists c I ą (cid:37) A ‹ ,I M ‹ ě c I . Combining these results for ‹ “ (cid:96) and ‹ “ r, one obtains a conjugate operator for thefree Hamiltonian M “ M (cid:96) ‘ M r introduced in Section 2.2. Namely, for any compact interval I Ă R z T M ,the operator A ,I : “ A (cid:96),I ‘ A r ,I satisfies the following properties :(a’) the operator A ,I is essentially self-adjoint on S p R , C q ‘ S p R , C q and on any set E ‘ E with E a core for Q ‹ , with closure denoted by the same symbol,(b’) the operator M is of class C p A ,I q ,(c’) there exists c I ą (cid:37) A ,I M ě c I . Remark 3.4.
What precedes implies in particular that the free Hamiltonian M has purely absolutelycontinuous spectrum except at the points of T M , where it may have eigenvalues. However, we alreadyknow from Proposition 2.8 that this does not occur. Therefore, σ p M q “ σ ac p M q “ σ ess p M q “ σ ess p M (cid:96) q Y σ ess p M r q . In this section, we show that the operator JA ,I J ˚ is a conjugate operator for the full Hamiltonian M introduced in Section 2.1. We start with the proof of the essential self-adjointness of JA ,I J ˚ in H w . Weuse the notation Q (see Remark 3.2) for the operator of multiplication by the variable in H w , p Qϕ qp x q : “ x ϕ p x q , ϕ P D p Q q : “ (cid:32) ϕ P H w | } Qϕ } H w ă 8 ( . roposition 3.5. For each compact interval I Ă R z T M the operator A I : “ JA ,I J ˚ is essentially self-adjoint on S p R , C q and on any other core for Q , with closure denoted by the same symbol.Proof. First, we observe that since J ˚ S p R , C q Ă D p Q q with Q : “ Q (cid:96) ‘ Q r the operator A I is well-defined and symmetric on S p R , C q Ă H w due to point (a’) above. Next, to prove the claim, we useNelson’s criterion of essential self-adjointness [22, Thm. X.37] applied to the triple ` A I , N, S p R , C q ˘ with N : “ Q ` S p R , C q is a core for N and that the operators Q ‹ Q ` , j ‹ , w ‹ w ´ j ‹ , Π ‹ ,n and U ´ ‹ p λ ,n U ‹ are bounded in H w . Moreover, we verify with direct calculations on S p R , C q that theoperators Π ‹ ,n and U ´ ‹ p λ ,n U ‹ belong to C p Q ‹ q (in H w ‹ ), and that their commutators r Π ‹ ,n , Q ‹ s and r U ´ ‹ p λ ,n U ‹ , Q ‹ s belong to C p Q q (in H w ). Then, a short computation using these properties gives thebound } A I ϕ } H w ď Const . } Nϕ } H w , ϕ P S p R , C q , and a slightly longer computation using the same properties shows that ˇˇ x A I ϕ, Nϕ y H w ´ x Nϕ, A I ϕ y H w ˇˇ ď Const . x Nϕ, ϕ y H w , ϕ P S p R , C q . Thus, the hypotheses of Nelson’s criterion are satisfied, and the claim follows.In order to prove that A I is a conjugate operator for M , we need two preliminary lemmas. Theyinvolve the two-Hilbert spaces difference of resolvents of M and M : B p z q : “ J p M ´ z q ´ ´ p M ´ z q ´ J, z P C z R . Lemma 3.6.
For each z P C z R , one has the inclusion B p z q P K p H , H w q .Proof. One has for p ϕ (cid:96) , ϕ r q P H B p z qp ϕ (cid:96) , ϕ r q “ ÿ ‹Pt (cid:96), r u ` j ‹ p M ‹ ´ z q ´ ´ p M ´ z q ´ j ‹ ˘ ϕ ‹ “ ÿ ‹Pt (cid:96), r u (cid:32)` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ ϕ ‹ ` “ j ‹ , p M ‹ ´ z q ´ ‰ ϕ ‹ ( . (3.9)Thus, an application of the standard result [30, Thm. 4.1] taking into account the properties of j ‹ impliesthat the operator “ j ‹ , p M ‹ ´ z q ´ ‰ is compact. This proves the claim for the second term in (3.9).For the first term in (3.9), we have the equalities ` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ “ p M ´ z q ´ p M ´ M ‹ q j ‹ p M ‹ ´ z q ´ ` p M ´ z q ´ p M ´ M ‹ q “ p M ‹ ´ z q ´ , j ‹ ‰ “ p M ´ z q ´ j ‹ p w ´ w ‹ q D p M ‹ ´ z q ´ ` p M ´ z q ´ p w ´ w ‹ qr D, j ‹ sp M ‹ ´ z q ´ ` p M ´ z q ´ p M ´ M ‹ q “ p M ‹ ´ z q ´ , j ‹ ‰ , (3.10)with j ‹ p w ´ w ‹ q and r D, j ‹ s matrix-valued functions vanishing at ˘8 . Thus, the operator in the first termin (3.9) is also compact, which concludes the proof. Lemma 3.7.
For each z P C z R and each compact interval I Ă R z T M , one has the inclusion B p z q A ,I æ D p A ,I q P K p H , H w q . Proof.
Since A ,I is essentially self-adjoint on S p R , C q ‘ S p R , C q , it is sufficient to show that B p z q A ,I æ ` S p R , C q ‘ S p R , C q ˘ P K p H , H w q . A ,I “ A (cid:96),I ‘ A r ,I , and each operator A ‹ ,I acts on S p R , C q as a sum Q ‹ F ‹ ,I ` G ‹ ,I ,with F ‹ ,I , G ‹ ,I bounded operators in H w ‹ mapping the set S p R , C q into D p Q ‹ q . These facts, togetherwith the compactness result of Lemma 3.6 and (3.9), imply that it is sufficient to show that ` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ Q ‹ æ S p R , C q P K p H w ‹ , H w q . and “ j ‹ , p M ‹ ´ z q ´ ‰ Q ‹ æ S p R , C q P K p H w ‹ , H w q . Now, if one takes Assumption 2.2 into account, the proof of these inclusions is similar to the proof ofLemma 3.6. We leave the details to the reader.Next, we will need the following theorem which is a direct consequence of Theorem 3.1 and Corollaries3.7-3.8 of [28].
Theorem 3.8.
Let H , A be self-adjoint operators in a Hilbert space H , let H be a self-adjoint operatorin a Hilbert space H , let J P B p H , H q , and let B p z q : “ J p H ´ z q ´ ´ p H ´ z q ´ J, z P C z R . Suppose there exists a set D Ă D p A J ˚ q such that JA J ˚ is essentially self-adjoint on D , with A itsself-adjoint extension. Finally, assume that(i) H is of class C p A q ,(ii) for each z P C z R , one has B p z q P K p H , H q ,(iii) for each z P C z R , one has B p z q A æ D p A q P K p H , H q ,(iv) for each η P C c p R q , one has η p H q p JJ ˚ ´ q η p H q P K p H q .Then, H is of class C p A q and r (cid:37) AH ě r (cid:37) A H . In particular, if A is conjugate to H at λ P R , then A isconjugate to H at λ . We are now ready to prove a Mourre estimate for M : Theorem 3.9 (Mourre estimate for M ) . Let I Ă R z T M be a compact interval. Then, M is of class C p A I q ,and r (cid:37) A I M p λ q ě r (cid:37) A ,I M p λ q “ min (cid:32)r (cid:37) A (cid:96),I M (cid:96) p λ q , r (cid:37) A r ,I M r p λ q ( ą for every λ P I .Proof. Theorem 3.3 and its restatement at the end of Section 3.2 give us the estimatemin (cid:32)r (cid:37) A (cid:96),I M (cid:96) p λ q , r (cid:37) A r ,I M r p λ q ( ą λ P I .In addition, the equality r (cid:37) A ,I M “ min (cid:32)r (cid:37) A (cid:96),I M (cid:96) , r (cid:37) A r ,I M r ( is a consequence of the definition of A ,I as a direct sumof A (cid:96),I and A r ,I (see [1, Prop. 8.3.5]).So, it only remains to show the inequality r (cid:37) A I M ě r (cid:37) A ,I M to prove the claim. For this, we apply Theorem3.8 with H “ M , H “ M and A “ A ,I , starting with the verification of its assumptions (i)-(iv) :the assumptions (i), (ii) and (iii) of Theorem 3.8 follow from point (b’) above, Lemma 3.6, and Lemma3.7, respectively. Furthermore, the assumption (iv) of Theorem 3.8 follows from the fact that for any η P C c p R q we have the inclusion η p M q p JJ ˚ ´ q η p M q “ η p M q ` w (cid:96) w ´ j (cid:96) ` w r w ´ j ´ ˘ p Q q η p M q P K p H w q , since w (cid:96) w ´ j (cid:96) ` w r w ´ j ´ “ p w (cid:96) ´ w q j (cid:96) w ´ ` p w r ´ w q j w ´ ` ` j (cid:96) ` j ´ ˘ is a matrix-valued function vanishing at ˘8 . These facts, together with Proposition 3.5 and the inclusion S p R , C q Ă D p A J ˚ q , imply that all the assumptions of Theorem 3.8 are satisfied. We thus obtain that r (cid:37) A I M ě r (cid:37) A ,I M , as desired. 16 .4 Spectral properties of the full Hamiltonian In this section, we determine the spectral properties of the full Hamiltonian M . We start by proving that M has the same essential spectrum as the free Hamiltonian M : Proposition 3.10.
One has σ ess p M q “ σ ess p M q “ σ p M (cid:96) q Y σ p M r q . To prove Proposition 3.10, we first need two preliminary lemmas. In the first lemma, we use thenotation χ Λ for the characteristic function of a Borel set Λ Ă R . Lemma 3.11. (a) The operator M is locally compact in H w , that is, χ B p Q qp M ´ i q ´ P K p H w q foreach bounded Borel set B Ă R .(b) Let ζ P C c ` R , r , ˘ satisfy ζ p x q “ for | x | ď and ζ p x q “ for | x | ě , and set ζ n p x q : “ ζ p x { n q for all x P R and n P N zt u . Then, lim n Ñ8 ›› r M, ζ n p Q qsp M ´ i q ´ ›› B p H w q “ . Moreover, the results of (a) and (b) also hold true for the operators M and Q in the Hilbert space H .Proof. (a) A direct computation shows that χ B p Q qp D ´ i q ´ “ ˆ i χ B p Q qp ` P q ´ χ B p Q q P p ` P q ´ χ B p Q q P p ` P q ´ i χ B p Q qp ` P q ´ ˙ , which implies that χ B p Q qp D ´ i q ´ is compact in L p R , C q since every entry of the matrix is compactin L p R q (see [30, Thm. 4.1]). Given that L p R , C q and H w have equivalent norms by Lemma 2.3(b), itfollows that χ B p Q qp D ´ i q ´ is also compact in H w . Finally, the resolvent identity (similar to (2.4)) p M ´ i q ´ “ p D ´ i q ´ w ´ ` i p D ´ i q ´ p w ´ ´ qp M ´ i q ´ shows that χ B p Q qp M ´ i q ´ is the sum of two compact operators in H w , and hence compact in H w . Thesame argument also shows that the operators M ‹ are locally compact in H w ‹ , and thus that M “ M (cid:96) ‘ M r is locally compact in H “ H w (cid:96) ‘ H w r .(b) Let ϕ P D p M q “ H p R , C q . Then, a direct computation taking into account the inclusion ζ n p Q q ϕ P D p M q gives r M, ζ n p Q qs ϕ “ w ˆ r P, ζ n p Q qsr P, ζ n p Q qs ˙ ϕ “ ´ in w ζ n p Q q ˆ ˙ ϕ. In consequence, we obtain that ›› r M, ζ n p Q qsp M ´ i q ´ ›› B p H w q ď Const . n ´ which proves the claim. Asbefore, the same argument also applies to the operators M ‹ in H w ‹ , and thus to the operator M “ M (cid:96) ‘ M r in H “ H w (cid:96) ‘ H w r .Lemma 3.11 is needed to prove that the essential spectra of M and M can be characterised interms of Zhislin sequences (see [12, Def. 10.4]). Zhislin sequences are particular types of Weyl sequencessupported at infinity as in the following lemma :
Lemma 3.12 (Zhislin sequences) . Let λ P R . Then, λ P σ ess p M q if and only if there exists a sequence t φ m u m P N zt u Ă D p M q , called Zhislin sequence, such that :(a) } φ m } H w “ for all m P N zt u ,(b) for each m P N zt u , one has φ m p x q “ if | x | ď m , c) lim m Ñ8 }p M ´ λ q φ m } H w “ .Similarly, λ P σ ess p M q if and only if there exists a sequence t φ m u m P N zt u Ă D p M q which meets theproperties (a), (b), (c) relative to the operator M .Proof. In view of Lemma 3.11, the claim can be proved by repeating step by step the arguments in theproof of [12, Thm. 10.6].We are now ready to complete the description of the essential spectrum of M : Proof of Proposition 3.10.
Take λ P σ ess p M q , let t φ m u m P N zt u Ă D p M q be an associated Zhislin sequence,and define for each m P N zt u φ m : “ c m p j (cid:96) φ m , j r φ m q P D p M q with c m : “ ››` j (cid:96) φ m , j r φ m ˘›› ´ H . Then, one has ›› φ m ›› H “ m P N zt u and φ m p x q “ | x | ď m . Furthermore, using successivelythe facts that j (cid:96) j r “
0, that p j (cid:96) ` j r q φ m “ φ m , and that 1 “ } φ m } H w “ x φ m , w ´ φ m y L p R , C q , one obtainsthat c ´ m “ @ φ m , ` w ´ (cid:96) j (cid:96) ` w ´ j r ˘ φ m D L p R , C q “ ` @ φ m , ` w ´ (cid:96) j (cid:96) ` w ´ j r ´ w ´ ˘ φ m D L p R , C q , which implies that lim m Ñ8 c m “ ›› p M ´ λ q φ m ›› H “ c m ÿ ‹Pt (cid:96), r u ›› p M ‹ ´ λ q j ‹ φ m ›› H w ‹ ď c m ÿ ‹Pt (cid:96), r u ´›› p M ´ λ q j ‹ φ m ›› H w ‹ ` ›› p M ‹ ´ M q j ‹ φ m ›› H w ‹ ¯ ď c m ÿ ‹Pt (cid:96), r u ´›› j ‹ p M ´ λ q φ m ›› H w ‹ ` ›› r M, j ‹ s φ m ›› H w ‹ ` ›› p w ‹ ´ w q D j ‹ φ m ›› H w ‹ ¯ . From the property (c) of Zhislin sequences, the boundedness of j ‹ , and the equivalence of the norms of H w ‹ and H w , one gets thatlim m Ñ8 ›› j ‹ p M ´ λ q φ m ›› H w ‹ ď Const . lim m Ñ8 ›› p M ´ λ q φ m ›› H w “ . Moreover, one has r M, j ‹ s φ m “ ´ iw j ` ˘ φ m , with j supported in r´ , s . This implies that r M, j ‹ s φ m “
0. For the same reason, one has Dj ‹ φ m “ j ‹ Dφ m , with the latter vector supported in x ď ´ m if ‹ “ (cid:96) and in x ě m if ‹ “ r. This, together with Assumption 2.2, implies that ›› p w ‹ ´ w q Dj ‹ φ m ›› H w ‹ ď Const . x m y ´ ´ ε } Dφ m } H w ‹ ď Const . x m y ´ ´ ε } Mφ m } H w . The last inequality, along with the equality lim m Ñ8 } Mφ m } H w “ | λ | (which follows from the property (c)of Zhislin sequences), gives lim m Ñ8 ›› p w ‹ ´ w q Dj ‹ φ m ›› H w ‹ “ . Putting all the pieces together, we obtain that lim m Ñ8 ›› p M ´ λ q φ m ›› H “
0. This concludes the proofthat t φ m u m P N zt u is a Zhislin sequence for the operator M and the point λ P σ ess p M q , and thus that σ ess p M q Ă σ ess p M q .For the opposite inclusion, take t φ m u m P N zt u Ă D p M q a Zhislin sequence for the operator M andthe point λ P σ ess p M q , and assume that λ P σ ess p M r q (if λ R σ ess p M r q , then λ P σ ess p M (cid:96) q and the sameproof applies if one replaces “right” with “left”). By extracting the nonzero right components from φ m tp , φ r m qu m P N zt u for M with t φ r m u m P N zt u Ă D p M r q a Zhislin sequence for M r . Then, we can construct as follows a new Zhislin sequence for M r with vectorssupported in r m, : Let ζ r P C c p R , r , sq satisfy ζ r p x q “ x ď ζ r p x q “ x ě
2, set ζ r m p x q : “ ζ r p x { m q for all x P R and m P N zt u , and choose n r m P N such that ›› χ r´ n r m , p Q r q φ r m ›› H w r Pp ´ { m, s . Next, define for each m P N zt u r φ r m : “ d m ζ r m p Q r q T k r m p r φ r m P D p M r q , with d m : “ ›› ζ r m p Q r q T k r m p r φ r m ›› ´ H w r , k r m P N such that k r m p r ě n r m ` m , and T k r m p r the operator of translationby k r m p r . One verifies easily that lim m Ñ8 d m “ r φ r m has support in r m, . Furthermore, sincethe operators M r and T k r m p r commute, one also has ›› p M r ´ λ q r φ r m ›› H w r ď d m ›› ζ r m p Q r q T k r m p r p M r ´ λ q φ rm ›› H w r ` d m ››“ M r , ζ r m p Q r q ‰ T k r m p r φ r m ›› H w r ď d m ›› p M r ´ λ q φ rm ›› H w r ` Const . m ´ ›› φ r m ›› H w r , which implies that lim m Ñ8 ›› p M r ´ λ q r φ r m ›› H w r “
0. Thus, t r φ r m u m P N zt u Ă D p M r q is a new Zhislin sequencefor M r with r φ r m supported in r m, .Now, define for each m P N zt u φ m : “ b m r φ r m P D p M q with b m : “ ›› r φ r m ›› ´ H w . Then, one has } φ m } H w “ m P N zt u and φ m p x q “ x ď m . Furthermore, using that1 “ ›› r φ r m ›› H w r “ x r φ r m , w ´ r φ r m y L p R , C q , one obtains that b ´ m : “ @ r φ r m , w ´ r φ r m D L p R , C q “ ` @ r φ r m , ` w ´ ´ w ´ ˘ r φ r m D L p R , C q , which implies that lim m Ñ8 b m “ ›› p M ´ λ q φ m ›› H w “ b m ›› p M ´ λ q r φ r m ›› H w ď b m ›› p M r ´ λ q r φ r m ›› H w ` b m ›› p w ´ w r q D r φ r m ›› H w . From the property (c) of Zhislin sequences and the equivalence of the norms of H w r and H w , one getsthat lim m Ñ8 ›› p M r ´ λ q r φ r m ›› H w ď Const . lim m Ñ8 ›› p M r ´ λ q r φ r m ›› H w r “ . Moreover, since D r φ r m is supported in r m, , one infers again from Assumption 2.2 that ›› p w ´ w r q D r φ r m ›› H w ď Const . x m y ´ ´ ε ›› D r φ r m ›› H w ď Const . x m y ´ ´ ε ›› M r r φ r m ›› H w r . The last inequality, along with the equality lim m Ñ8 ›› M r r φ r m ›› H w r “ | λ | , giveslim m Ñ8 ›› p w ´ w r q D r φ r m ›› H w “ . Putting all the pieces together, we obtain that lim m Ñ8 ›› p M ´ λ q φ m ›› H w “
0. This concludes the proofthat t φ m u m P N zt u is a Zhislin sequence for the operator M and the point λ P σ ess p M q , and thus that σ ess p M q Ă σ ess p M q . In consequence, we obtained that σ ess p M q “ σ ess p M q , which completes the proofin view of Remark 3.4. 19n order to determine more precise spectral properties of M , we now prove that for each compactinterval I Ă R z T M the Hamiltonian M is of class C ` ε p A I q for some ε P p , q , which is a regularitycondition slightly stronger than the condition M of class C p A I q already established in Theorem 3.9. Westart by giving a convenient formula for the commutator rp M ´ z q ´ , A I s , z P C z R , in the form sense on S p R , C q : “ p M ´ z q ´ , A I ‰ “ ` p M ´ z q ´ J ´ J p M ´ z q ´ ˘ A ,I J ˚ ´ JA ,I ` J ˚ p M ´ z q ´ ´ p M ´ z q ´ J ˚ ˘ ` J “ p M ´ z q ´ , A ,I ‰ J ˚ “ ÿ ‹Pt (cid:96), r u (cid:32)` p M ´ z q ´ ´ p M ‹ ´ z q ´ ˘ j ‹ A ‹ ,I Z ‹ ` “ p M ‹ ´ z q ´ , j ‹ ‰ A ‹ ,I Z ‹ ´ j ‹ A ‹ ,I Z ‹ ` p M ´ z q ´ ´ p M ‹ ´ z q ´ ˘ ` j ‹ A ‹ ,I “ p M ‹ ´ z q ´ , Z ‹ ‰ ` j ‹ “ p M ‹ ´ z q ´ , A ‹ ,I ‰ Z ‹ ( “ ÿ ‹Pt (cid:96), r u ` C ‹ ` j ‹ “ p M ‹ ´ z q ´ , A ‹ ,I ‰ Z ‹ ˘ with Z ‹ : “ w ‹ w ´ j ‹ “ j ‹ ´ p w ´ w ‹ q j ‹ w ´ , and C ‹ : “ ` p M ´ z q ´ ´ p M ‹ ´ z q ´ ˘ j ‹ A ‹ ,I Z ‹ ` “ p M ‹ ´ z q ´ , j ‹ ‰ A ‹ ,I Z ‹ ´ j ‹ A ‹ ,I Z ‹ ` p M ´ z q ´ ´ p M ‹ ´ z q ´ ˘ ` j ‹ A ‹ ,I “ p M ‹ ´ z q ´ , Z ‹ ‰ . As already shown in the previous section, all the terms in C ‹ extend to bounded operators, and we keepthe same notation for these extensions.In order to show that p M ´ z q ´ P C ` ε p A I q , it is enough to prove that j ‹ “ p M ‹ ´ z q ´ , A ‹ ,I ‰ Z ‹ P C p A I q and to check that ›› e ´ itA I C ‹ e itA I ´ C ‹ ›› B p H w q ď Const . t ε for all t P p , q . (3.11)Since the first proof reduces to computations similar to the ones presented in the previous section, we shallconcentrate on the proof of (3.11). First of all, algebraic manipulations as presented in [1, pp. 325-326]or [26, Sec. 4.3] show that for all t P p , q ›› e ´ itA I C ‹ e itA I ´ C ‹ ›› B p H w q ď Const . ´›› sin p tA I q C ‹ ›› B p H w q ` ›› sin p tA I qp C ‹ q ˚ ›› B p H w q ¯ ď Const . ´›› tA I p tA I ` i q ´ C ‹ ›› B p H w q ` ›› tA I p tA I ` i q ´ p C ‹ q ˚ ›› B p H w q ¯ . Furthermore, if we set A t : “ tA I p tA I ` i q ´ and Λ t : “ t x Q yp t x Q y ` i q ´ , we obtain that A t “ ` A t ` i p tA I ` i q ´ A I x Q y ´ ˘ Λ t with A I x Q y ´ P B p H w q due to (3.7). Thus, since ›› A t ` i p tA I ` i q ´ A I x Q y ´ ›› B p H w q is bounded by aconstant independent of t P p , q , it is sufficient to prove that } Λ t C ‹ } B p H w q ` } Λ t p C ‹ q ˚ } B p H w q ď Const . t ε for all t P p , q ,and to prove this estimate it is sufficient to show that the operators x Q y ε C ‹ and x Q y ε p C ‹ q ˚ definedin the form sense on S p R , C q extend continuously to elements of B p H w q . Finally, some lengthy butstraightforward computations show that these two last conditions are implied by the following two lemmas : Lemma 3.13. M ‹ is of class C px Q y α q for each ‹ P t (cid:96), r u and α P r , s . roof. One can verify directly that the unitary group generated by the operator x Q y α leaves the domain D p M ‹ q “ H p R , C q invariant and that the commutator r M ‹ , x Q y α s defined in the form sense on S p R , C q extends continuously to a bounded operator. Since the set S p R , C q is a core for M ‹ , these propertiestogether with [1, Thm. 6.3.4(a)] imply the claim. Lemma 3.14.
One has j ‹ P C p A ‹ ,I q for each ‹ P t (cid:96), r u and each compact interval I Ă R z T M .Proof. By using the commutator expansions [1, Thm. 5.5.3] and (3.7), one gets the following equalitiesin form sense on S p R , C q : “ j ‹ , A ‹ ,I ‰ “ ? π ż d τ ż R d x e iτxQ ‹ r Q ‹ , A ‹ ,I s e i p ´ τ q xQ ‹ p j p x q“ ? π ÿ n P N p I q ż d τ ż R d x e iτxQ ‹ “ Q ‹ , Π ‹ ,n `q λ ,n Q ‹ ` Q ‹ q λ ,n ˘ Π ‹ ,n ‰ e i p ´ τ q xQ ‹ p j p x q“ ? π ÿ n P N p I q ż d τ ż R d x e iτxQ ‹ !“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰ Q ‹ ` Q ‹ “ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰ ` “ Q ‹ , Π ‹ ,n q λ ‹ ,n r Q ‹ , Π ‹ ,n s ‰ ´ “ Q ‹ , r Q ‹ , Π ‹ ,n s q λ ‹ ,n Π ‹ ,n ‰) e i p ´ τ q xQ ‹ p j p x q with q λ ,n : “ U ´ ‹ p λ ,n U ‹ and with each commutator in the last equality extending continuously to abounded operator. Since p j is integrable, the last two terms give bounded contributions. Furthermore, thefirst two terms can be rewritten as f `“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰˘ Q ‹ ` Q ‹ f `“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰˘ with f : B p H w ‹ q Ñ B p H w ‹ q , B ÞÑ f p B q : “ ? π ÿ n P N p I q ż d τ ż R d x e iτxQ ‹ B e i p ´ τ q xQ ‹ p j p x q . But, since “ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰ P C k p Q ‹ q for each k P N , and since p j is a Schwartz function, one infersfrom [1, Thm. 5.5.3] that the operator f `“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰˘ is regularising in the Besov scale asso-ciated to the operator Q ‹ . This implies in particular that the operators f `“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰˘ Q ‹ and Q ‹ f `“ Q ‹ , Π ‹ ,n q λ ,n Π ‹ ,n ‰˘ extend continuously to bounded operators, as desired.We can now give in the next theorem a description of the structure of the spectrum of the fullHamiltonian M . The next theorem also shows that the set T M can be interpreted as the set of thresholdsin the spectrum of M : Theorem 3.15.
In any compact interval I Ă R z T M , the operator M has at most finitely many eigenvalues,each one of finite multiplicity, and no singular continuous spectrum.Proof. The computations at the beginning of this section together with Lemmas 3.13 & 3.14 implythat M is of class C ` ε p A I q for some ε P p , q , and Theorem 3.9 implies that the condition (3.5) ofTheorem 3.1 is satisfied on I . So, one can apply Theorem 3.1(b) to conclude. We discuss in this section the existence and the completeness, under smooth perturbations, of the localwave operators for self-adjoint operators in a two-Hilbert spaces setting. Namely, given two self-adjoint21perators H , H in Hilbert spaces H , H with spectral measures E H , E H , an identification operator J P B p H , H q , and an open set I Ă R , we recall criteria for the existence and the completeness of the stronglimits W ˘ p H, H , J, I q : “ s-lim t Ñ˘8 e itH J e ´ itH E H p I q under the assumption that the two-Hilbert spaces difference of resolvents J p H ´ z q ´ ´ p H ´ z q ´ J, z P C z R , factorises as a product of a locally H -smooth operator on I and a locally H -smooth operator on I .We start by recalling some facts related to the notion of J -completeness. Let N ˘ p H, J, I q be thesubsets of H defined by N ˘ p H, J, I q : “ " ϕ P H | lim t Ñ˘8 ›› J ˚ e ´ itH E H p I q ϕ ›› H “ * . Then, it is clear that N ˘ p H, J, I q are closed subspaces of H and that E H p R z I q H Ă N ˘ p H, J, I q , and it isshown in [38, Sec. 3.2] that H is reduced by N ˘ p H, J, I q and thatRan ` W ˘ p H, H , J, I q ˘ K N ˘ p H, J, I q . In particular, one has the inclusionRan ` W ˘ p H, H , J, I q ˘ Ă E H p I q H a N ˘ p H, J, I q , which motivates the following definition : Definition 4.1 ( J -completeness) . Assume that the local wave operators W ˘ p H, H , J, I q exist. Then, W ˘ p H, H , J, I q are J -complete on I if Ran ` W ˘ p H, H , J, I q ˘ “ E H p I q H a N ˘ p H, J, I q . Remark 4.2.
In the particular case H “ H and J “ H , the J -completeness on I reduces to the complete-ness of the local wave operators W ˘ p H, H , J, I q on I in the usual sense. Namely, Ran ` W ˘ p H, H , H , I q ˘ “ E H p I q H , and the operators W ˘ p H, H , H , I q are unitary from E H p I q H to E H p I q H . The following criterion for J -completeness has been established in [38, Thm. 3.2.4] : Lemma 4.3.
If the local wave operators W ˘ p H, H , J, I q and W ˘ p H , H, J ˚ , I q exist, then W ˘ p H, H , J, I q are J -complete on I . For the next theorem, we recall that the spectral support supp H p ϕ q of a vector ϕ P H with respectto H is the smallest closed set I Ă R such that E H p I q ϕ “ ϕ . Theorem 4.4.
Let H , H be self-adjoint operators in Hilbert spaces H , H with spectral measures E H , E H , J P B p H , H q , I Ă R an open set, and G an auxiliary Hilbert space. For each z P C z R , assume there exist T p z q P B p H , G q locally H -smooth on I and T p z q P B p H , G q locally H -smooth I such that J p H ´ z q ´ ´ p H ´ z q ´ J “ T p z q ˚ T p z q . Then, the local wave operators W ˘ p H, H , J, I q “ s-lim t Ñ˘8 e itH J e ´ itH E H p I q (4.1) exist, are J -complete on I , and satisfy the relations W ˘ p H, H , J, I q ˚ “ W ˘ p H , H, J ˚ , I q and W ˘ p H, H , J, I q η p H q “ η p H q W ˘ p H, H , J, I q for each bounded Borel function η : R Ñ C . roof. We adapt the proof of [1, Thm. 7.1.4] to the two-Hilbert spaces setting. The existence of thelimits (4.1) is a direct consequence of the following claims : for each ϕ P H with I : “ supp H p ϕ q Ă I compact, and for each η P C c p I q such that η ” I , we have thats-lim t Ñ˘8 η p H q e itH J e ´ itH ϕ exist and lim t Ñ˘8 ››` ´ η p H q ˘ e itH J e ´ itH ϕ ›› H “ . (4.2)To prove the first claim in (4.2), take ϕ P H and t P R , and observe that the operators W p t q : “ η p H q e itH J e ´ itH satisfy for z P C z R and s ď t ˇˇ@ p H ´ z q ´ ϕ, ` W p t q ´ W p s q ˘ p H ´ z q ´ ϕ D H ˇˇ “ ˇˇˇˇż ts d u dd u @ e ´ iuH η p H q ϕ, p H ´ z q ´ J p H ´ z q ´ e ´ iuH ϕ D H ˇˇˇˇ “ ˇˇˇˇż ts d u @ e ´ iuH η p H q ϕ, p H ´ z q ´ p HJ ´ JH qp H ´ z q ´ e ´ iuH ϕ D H ˇˇˇˇ “ ˇˇˇˇż ts d u @ e ´ iuH η p H q ϕ, ` J p H ´ z q ´ ´ p H ´ z q ´ J ˘ e ´ iuH ϕ D H ˇˇˇˇ “ ˇˇˇˇż ts d u @ T p z q e ´ iuH η p H q ϕ, T p z q e ´ iuH ϕ D G ˇˇˇˇ ď ˆż ts d u ›› T p z q e ´ iuH η p H q ϕ ›› G ˙ { ˆż ts d u ›› T p z q e ´ iuH ϕ ›› G ˙ { ď c { I } ϕ } H ˆż ts d u ›› T p z q e ´ iuH ϕ ›› G ˙ { , with I : “ supp p η q and c I the constant appearing in the definition (3.4) of a locally H -smooth operator.Since the set p H ´ z q ´ H is dense in H and T is locally H -smooth on I , it follows that ››` W p t q ´ W p s q ˘ p H ´ z q ´ ϕ ›› H Ñ s Ñ 8 or t Ñ ´8 . Applying this result with ϕ replaced by p H ´ z q ϕ we infer that ››` W p t q ´ W p s q ˘ ϕ ›› H “ ››` W p t q ´ W p s q ˘ p H ´ z q ´ p H ´ z q ϕ ›› H Ñ s Ñ 8 or t Ñ ´8 , which proves the first claim in (4.2).To prove the second claim in (4.2), we take η P C c p I q such that η ” I and η η “ η . Then,we have ϕ “ η p H q ϕ and ` ´ η p H q ˘ J η p H q “ ` ´ η p H q ˘` J η p H q ´ η p H q J ˘ , and thus the second claim in (4.2) follows fromlim t Ñ˘8 ››`
J η p H q ´ η p H q J ˘ e ´ itH ϕ ›› H “ . Since the vector space generated by the functions R Q x ÞÑ p x ´ z q ´ P C , z P C z R , is dense in C p R q ,it is sufficient to show thatlim t Ñ˘8 ››` J p H ´ z q ´ ´ p H ´ z q ´ J ˘ e ´ itH ϕ ›› H “ , z P C z R . Now, we have for every ϕ P H ˇˇ@ ϕ, ` J p H ´ z q ´ ´ p H ´ z q ´ J ˘ e ´ itH ϕ D H ˇˇ “ ˇˇ@ T p z q ϕ, T p z q e ´ itH ϕ D G ˇˇ ď ›› T p z q ϕ ›› G ›› T p z q e ´ itH ϕ ›› G . ›› T p z q e ´ itH ϕ ›› G Ñ | t | Ñ 8 . But since T p z q e ´ itH ϕ andits derivative are square integrable in t , this follows from a standard Sobolev embedding argument. So,the existence of the limits (4.1) has been established. Similar arguments, using the relation p H ´ z q ´ J ˚ ´ J ˚ p H ´ z q ´ “ T p z q ˚ T p z q instead of J p H ´ z q ´ ´ p H ´ z q ´ J “ T p z q ˚ T p z q , show that W ˘ p H , H, J ˚ , I q exists too. This, together with standard arguments in scattering theory, impliesthe claims that follow (4.1).As a consequence of Theorem 3.1(a) & Theorem 4.4, we obtain the following criterion for theexistence and completeness of the local wave operators : Corollary 4.5.
Let H , H be self-adjoint operators in Hilbert spaces H , H with spectral measures E H , E H and A , A self-adjoint operators in H , H . Assume that H , H are of class C ` ε p A q , C ` ε p A q for some ε P p , q . Let I : “ (cid:32)r µ A p H qz σ p p H q ( X (cid:32)r µ A p H qz σ p p H q ( ,J P B p H , H q , G an auxiliary Hilbert space, and for each z P C z R suppose there exist T p z q P B p H , G q and T p z q P B p H , G q with J p H ´ z q ´ ´ p H ´ z q ´ J “ T p z q ˚ T p z q (4.3) and such that T p z q extends continuously to an element of B ` D px A y s q ˚ , G ˘ and T p z q extends continu-ously to an element of B ` D px A y s q ˚ , G ˘ for some s ą { . Then, the local wave operators W ˘ p H, H , J, I q “ s-lim t Ñ˘8 e itH J e ´ itH E H p I q exist, are J -complete on I , and satisfy the relations W ˘ p H, H , J, I q ˚ “ W ˘ p H , H, J ˚ , I q and W ˘ p H, H , J, I q η p H q “ η p H q W ˘ p H, H , J, I q for each bounded Borel function η : R Ñ C . In the case of the pair t M , M u , we obtain the following result on the existence and completeness ofthe wave operators; we use the notation E M ac for the orthogonal projection on the absolutely continuoussubspace of M : Theorem 4.6.
Let I max : “ σ p M qzt T M Y σ p p M qu . Then, the local wave operators W ˘ p M, M , J, I max q : “ s-lim t Ñ˘8 e itM J e ´ itM E M p I max q exist and satisfy Ran ` W ˘ p M, M , J, I max q ˘ “ E M ac H w . In addition, the relations W ˘ p M, M , J, I max q ˚ “ W ˘ p M , M, J ˚ , I max q and W ˘ p M, M , J, I max q η p M q “ η p M q W ˘ p M, M , J, I max q hold for each bounded Borel function η : R Ñ C . roof. All the claims except the equality Ran ` W ˘ p M, M , J, I max q ˘ “ E M ac H w follow from Corollary 4.5whose assumptions are verified below.Let I Ă σ p M qzt T M Y σ p p M qu be a compact interval. Then, we know from Section 3.2 that M is ofclass C p A ,I q and from Section 3.4 that M is of class C ` ε p A I q for some ε P p , q . Moreover, Theorems3.3 & 3.9 imply that I Ă r µ A ,I p M q X (cid:32)r µ A I p M qz σ p p M q ( . Therefore, in order to apply Corollary 4.5, it is sufficient to prove that for any z P C z R the operator B p z q “ J p M ´ z q ´ ´ p M ´ z q ´ J factorises as a product of two locally smooth operators as in (4.3). For that purpose, we set s : “ ` r ε with r ε P p , ε q , we define D : “ (cid:32) S p R , C q ‘ S p R , C q ( ˆ S p R , C q Ă H ˆ H w , and we consider the sesquilinear form D Q ` p ϕ (cid:96) , ϕ r q , ϕ ˘ ÞÑ @ x Q y s ϕ, B p z q ` x Q (cid:96) y s ϕ (cid:96) , x Q r y s ϕ r ˘D H w P C . (4.4)Our first goal is to show that this sesquilinear form is continuous for the topology of H ˆ H w . However,since the necessary computations are similar to the ones presented in Sections 3.3-3.4, we only sketchthem. We know from (3.9) that B p z q ` x Q (cid:96) y s ϕ (cid:96) , x Q r y s ϕ r ˘ “ ÿ ‹Pt (cid:96), r u (cid:32)` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ x Q ‹ y s ϕ ‹ ` “ j ‹ , p M ‹ ´ z q ´ ‰ x Q ‹ y s ϕ ‹ ( . So, we have to establish the continuity of the sesquilinear forms S p R , C q ˆ S p R , C q Q p ϕ ‹ , ϕ q ÞÑ @ x Q y s ϕ, ` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ x Q ‹ y s ϕ ‹ D H w P C (4.5)and S p R , C q ˆ S p R , C q Q p ϕ ‹ , ϕ q ÞÑ @ x Q y s ϕ, “ j ‹ , p M ‹ ´ z q ´ ‰ x Q ‹ y s ϕ ‹ D H w P C . (4.6)For the first one, we know from (3.10) that ` p M ‹ ´ z q ´ ´ p M ´ z q ´ ˘ j ‹ x Q ‹ y s “ p M ´ z q ´ j ‹ p w ´ w ‹ q D p M ‹ ´ z q ´ x Q ‹ y s ` p M ´ z q ´ p w ´ w ‹ qr D, j ‹ sp M ‹ ´ z q ´ x Q ‹ y s ` p M ´ z q ´ p M ´ M ‹ q “ p M ‹ ´ z q ´ , j ‹ ‰ x Q ‹ y s . (4.7)By inserting this expression into (4.5), by taking the C px Q y α q -property of M and M ‹ into account,and by observing that the operators r D, x Q ‹ y s s , x Q y s j ‹ p w ´ w ‹ qx Q ‹ y s and x Q y s r D, j ‹ sx Q ‹ y s defined on S p R , C q extend continuously to elements of B p H w q , one obtains that the sesquilinear forms definedby the first two terms in (4.7) are continuous for the topology of H w ‹ ˆ H w . The sesquilinear formdefined by the third term in (4.7) and the sesquilinear form (4.6) can be treated simultaneously. Indeed,the factor r j ‹ , p M ‹ ´ z q ´ s can be computed explicitly and contains a factor j which has compactsupport. Therefore, since x Q y s j x Q ‹ y s P B p H w q , a few more commutator computations show that thetwo remaining sesquilinear forms are continuous for the topology of H w ‹ ˆ H w .In consequence, the sesquilinear form (4.4) is continuous for the topology of H ˆ H w , and thuscorresponds to a bounded operator F z P B p H , H w q . Therefore, if we set T p z q : “ x Q (cid:96) y ´ s ‘ x Q r y ´ s P B p H q and T p z q : “ F ˚ z x Q y ´ s P B p H w , H q , we obtain that B p z q “ T p z q ˚ T p z q . On another hand, we know from computations presented in Sec-tion 3.4 that x Q y ´ s P B ` D px Q y s q ˚ , H w ˘ Ă B ` D px A I y s q ˚ , H w ˘ , x Q (cid:96) y ´ s ‘ x Q r y ´ s P B ` D px Q (cid:96) y s ‘ x Q r y s q ˚ , H ˘ Ă B ` D px A ,I y s q ˚ , H ˘ . So, we have the inclusions T p z q P B ` D px A I y s q ˚ , H ˘ and T p z q P B ` D px A ,I y s q ˚ , H ˘ , and thus all the assumptions of Corollary 4.5 are verified.Hence it only remains to show that Ran ` W ˘ p M, M , J, I max q ˘ “ E M ac H w . For that purpose, we firstrecall from the proof of Theorem 3.9 that E M p I qp JJ ˚ ´ q E M p I q P K p H w q . Then, since M has purelyabsolutely continuous spectrum in I one infers from the RAGE theorem thats-lim t Ñ˘8 E M p I q e itM p JJ ˚ ´ q e ´ itM E M p I q “ , and consequently that N ˘ p M, J, I q “ E M p R z I q H w . By using the J -completeness on I of the local waveoperators and that M has purely absolutely continuous spectrum in I , we thus obtainRan ` W ˘ p M, M , J, I q ˘ “ E M p I q H a N ˘ p M, J, I q “ E M p I q H w “ E M p I q H w X E M ac H w . By putting together these results for different intervals I and by using Proposition 3.10, we thus get thatRan ` W ˘ p M, M , J, I max q ˘ “ E M p I max q H w X E M ac H w “ E M ` σ ess p M q ˘ H w X E M ac H w “ E M ac H w , which concludes the proof. Remark 4.7.
Let I Ă σ p M qzt T M Y σ p p M qu be a compact interval and let p ϕ (cid:96) , ϕ r q P H . Then, we have W ˘ p M, M , J, I qp ϕ (cid:96) , ϕ r q “ s-lim t Ñ˘8 e itM J e ´ itM E M p I qp ϕ (cid:96) , ϕ r q“ s-lim t Ñ˘8 e itM ` J (cid:96) e ´ itM (cid:96) E M (cid:96) p I q ϕ (cid:96) ` J r e ´ itM r E M r p I q ϕ r ˘ “ W ˘ p M, M (cid:96) , J (cid:96) , I q ϕ (cid:96) ` W ˘ p M, M r , J r , I q ϕ r with W ˘ p M, M ‹ , J ‹ , I q : “ s-lim t Ñ˘8 e itM J ‹ e ´ itM ‹ E M ‹ p I q (4.8) and J ‹ P B p H w ‹ , H w q given by J ‹ ϕ ‹ : “ j ‹ ϕ ‹ , ϕ ‹ P H w ‹ . (4.9) That is, the operators W ˘ p M, M , J, I q act as the sum of the local wave operators W ˘ p M, M ‹ , J ‹ , I q : W ˘ p M, M , J, I qp ϕ (cid:96) , ϕ r q “ W ˘ p M, M (cid:96) , J (cid:96) , I q ϕ (cid:96) ` W ˘ p M, M r , J r , I q ϕ r . In order to get a better understanding of the initial sets of the isometries W ˘ p M, M , J, I max q somepreliminary considerations on the asymptotic velocity operators for M (cid:96) and M r are necessary. First, wedefine for each ‹ P t (cid:96), r u and n P N the spaces H ‹ ,n : “ p Π ‹ ,n H τ, ‹ and H ,n : “ p Π ‹ ,n (cid:32) H τ, ‹ X C p R , h ‹ q ( , and note that H τ, ‹ decomposes into the internal direct sum H τ, ‹ “ ‘ n P N H ‹ ,n and that the operator x M ‹ isreduced by this decomposition, namely, x M ‹ “ ř n P N p λ ‹ ,n p Π ‹ ,n . Next, we introduce the self-adjoint operator p V ‹ in H τ, ‹ p V ‹ : “ ÿ n P N p λ ,n p Π ‹ ,n , D `p V ‹ ˘ : “ u P H τ, ‹ | ››p V ‹ u ›› H τ, ‹ “ ÿ n P N ››p λ ,n p Π ‹ ,n u ›› H τ, ‹ ă 8 + . V ‹ for M ‹ in H w ‹ as V ‹ : “ U ´ ‹ p V ‹ U ‹ , D p V ‹ q : “ U ´ ‹ D `p V ‹ ˘ , and the asymptotic velocity operator V for M in H as the direct sum V : “ V (cid:96) ‘ V r . Additionally, we define the family of self-adjoint operators in H w ‹ Q ‹ p t q : “ e itM ‹ Q ‹ e ´ itM ‹ , t P R , D ` Q ‹ p t q ˘ : “ e itM ‹ D p Q ‹ q , and the corresponding family of self-adjoint operators in H Q p t q : “ Q (cid:96) p t q ‘ Q r p t q , t P R . Our next result is inspired by the result of [32, Thm. 4.1] in the setup of quantum walks. In theproof, we use the linear span H fin ‹ ,τ of elements of H ,n : H fin ‹ ,τ : “ n P N u n P à n P N H ,n | u n ‰ n + . Proposition 4.8.
For each ‹ P t (cid:96), r u and z P C z R , we have s-lim t Ñ˘8 ´ Q ‹ p t q t ´ z ¯ ´ “ ` V ‹ ´ z ˘ ´ . Proof.
For each t P R , we have the inclusion U ´ ‹ H fin ‹ ,τ Ă (cid:32) D p V ‹ q X D ` Q ‹ p t q ˘( . Furthermore, if u P H fin ‹ ,τ ,then we have ` V ‹ ´ z ˘ ´ U ´ ‹ u “ U ´ ‹ `p V ‹ ´ z ˘ ´ u P U ´ ‹ H fin ‹ ,τ . As a consequence, the following equality holds for all t P R zt u and u P H fin ‹ ,τ : ˆ´ Q ‹ p t q t ´ z ¯ ´ ´ ` V ‹ ´ z ˘ ´ ˙ U ´ ‹ u “ ´ Q ‹ p t q t ´ z ¯ ´ ´ V ‹ ´ Q ‹ p t q t ¯ ` V ‹ ´ z ˘ ´ U ´ ‹ u. Since ››` Q ‹ p t q t ´ z ˘ ´ ›› B p H w ‹ q ď | Im p z q| ´ and ›› p V ‹ ´ z q ´ ›› B p H w ‹ q ď | Im p z q| ´ , and since U ´ ‹ H fin ‹ ,τ isdense in H w ‹ , it follows that it is sufficient to prove thatlim t Ñ˘8 ›››´ V ‹ ´ Q ‹ p t q t ¯ ϕ ‹ ››› H w ‹ “ ϕ ‹ P U ´ ‹ H fin ‹ ,τ .Now, a direct calculation using the Bloch-Floquet transform gives for ϕ ‹ “ U ´ ‹ u with u P H fin ‹ ,τ ›››´ V ‹ ´ Q ‹ p t q t ¯ ϕ ‹ ››› H w ‹ “ ż Y ˚‹ d k ››››› ÿ n P N `p λ ,n p Π ‹ ,n u ˘ p k q ´ ˜ e it x M ‹ x Q ‹ t ÿ n P N e ´ it p λ ‹ ,n p Π ‹ ,n u ¸ p k q ››››› h ‹ “ t ż Y ˚‹ d k ››››› ÿ n P N ` x Q ‹ p Π ‹ ,n u ˘ p k q ››››› h ‹ , where in the last equation we have used that x Q ‹ acts as i B k in H τ, ‹ . Since u P H fin ‹ ,τ , the summation over n P N is finite, and since the map Y ˚‹ Q k ÞÑ ` x Q ‹ p Π ‹ ,n u ˘ p k q P h ‹ is bounded, one deduces that ›››´ V ‹ ´ Q ‹ p t q t ¯ ϕ ‹ ››› H w ‹ “ O ` t ´ ˘ , which implies the claim. 27n the next proposition, we determine the initial sets of the isometries W ˘ p M, M ‹ , J ‹ , I q : H w ‹ Ñ H w defined in (4.8). In the statement, we use the fact that the operators M ‹ and V ‹ strongly commute. Wealso use the notations χ ` and χ ´ for the characteristic functions of the intervals p , and p´8 , q ,respectively. Proposition 4.9. (a) Let I Ă σ p M (cid:96) qz T (cid:96) be a compact interval, then the operators W ˘ p M, M (cid:96) , J (cid:96) , I q : H w (cid:96) Ñ H w are partial isometries with initial sets χ ¯ p V (cid:96) q E M (cid:96) p I q H w (cid:96) .(b) Let I Ă σ p M r qz T r be a compact interval, then the operators W ˘ p M, M r , J r , I q : H w r Ñ H w arepartial isometries with initial sets χ ˘ p V r q E M r p I q H w r . Before the proof, let us observe that if I Ă σ p M ‹ qz T ‹ is a compact interval, then we have theequalities χ ´ p V ‹ q E M ‹ p I q “ χ p´8 , s p V ‹ q E M ‹ p I q and χ ` p V ‹ q E M ‹ p I q “ χ r , `8q p V ‹ q E M ‹ p I q (4.10)due to the definition of the set T ‹ . Proof.
Our proof is inspired by the proof of [27, Prop. 3.4]. We first show the claim for W ` p M, M (cid:96) , J (cid:96) , I q .So, let ϕ (cid:96) P H w (cid:96) . If ϕ (cid:96) K E M (cid:96) p I q H w (cid:96) , then ϕ (cid:96) P ker ` W ` p M, M (cid:96) , J (cid:96) , I q ˘ . Thus, we can assume that ϕ (cid:96) P E M (cid:96) p I q H w (cid:96) . Next, let us show that if ϕ (cid:96) P χ ` p V (cid:96) q H w (cid:96) then again ϕ (cid:96) P ker ` W ` p M, M (cid:96) , J (cid:96) , I q ˘ . For this,assume that χ r ε, p V (cid:96) q ϕ (cid:96) “ ϕ (cid:96) for some ε ą
0. Then, it follows from (4.8)-(4.9) that ›› W ` p M, M (cid:96) , J (cid:96) , I q ϕ (cid:96) ›› H w “ ›››› s-lim t Ñ`8 e itM J (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›››› H w “ lim t Ñ`8 ›› e itM J (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›› H w ď Const . lim t Ñ`8 ›› e itM (cid:96) j (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›› H w(cid:96) . Now, let η (cid:96) P C p R , r , sq satisfy η (cid:96) p x q “ x ă η (cid:96) p x q “ x ě ε . Then, one has for each t ą ›› e itM (cid:96) j (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›› H w(cid:96) ď ›› e itM (cid:96) η (cid:96) p Q (cid:96) { t q e ´ itM (cid:96) ϕ (cid:96) ›› H w(cid:96) . Furthermore, since η (cid:96) p V (cid:96) q ϕ (cid:96) “ η (cid:96) p V (cid:96) q χ r ε, p V (cid:96) q ϕ (cid:96) “
0, one infers from Proposition 4.8 and from astandard result on strong resolvent convergence [25, Thm. VIII.20(b)] thatlim t Ñ`8 ›› e itM (cid:96) η (cid:96) p Q (cid:96) { t q e ´ itM (cid:96) ϕ (cid:96) ›› H w(cid:96) “ ›› η (cid:96) p V (cid:96) q ϕ (cid:96) ›› H w(cid:96) “ . Putting together what precedes, one obtains that ϕ (cid:96) “ χ r ε, p V (cid:96) q ϕ (cid:96) P ker ` W ` p M, M (cid:96) , J (cid:96) , I q ˘ , and thena density argument taking into account the second equation in (4.10) implies that χ ` p V (cid:96) q H w (cid:96) Ă ker ` W ` p M, M (cid:96) , J (cid:96) , I q ˘ . To show that W ` p M, M (cid:96) , J (cid:96) , I q is an isometry on χ ´ p V (cid:96) q E M (cid:96) p I q H w (cid:96) , take ϕ (cid:96) P χ ´ p V (cid:96) q E M (cid:96) p I q H w (cid:96) with χ p´8 , ´ ε s p V (cid:96) q ϕ (cid:96) “ ϕ (cid:96) for some ε ą
0, and let ζ (cid:96) P C p R , r , sq satisfy ζ (cid:96) p x q “ x ď ´ ε and ζ (cid:96) p x q “ x ą ´ ε {
2. Then, using successively the identity E M (cid:96) p I q ϕ (cid:96) “ ϕ (cid:96) , the unitarity of e itM in H w and of e ´ itM (cid:96) in H w (cid:96) , the definition (4.9) of J (cid:96) , the definition of V (cid:96) , and the fact that χ p´8 , ´ ε s p V (cid:96) q ϕ (cid:96) “ ϕ (cid:96) , one gets ˇˇˇ›› W ` p M, M (cid:96) , J (cid:96) , I q ϕ (cid:96) ›› H w ´ } ϕ (cid:96) } H w(cid:96) ˇˇˇ “ lim t Ñ`8 ˇˇˇ›› e itM J (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›› H w ´ } ϕ (cid:96) } H w(cid:96) ˇˇˇ “ lim t Ñ`8 ˇˇˇ›› J (cid:96) e ´ itM (cid:96) ϕ (cid:96) ›› H w ´ ›› e ´ itM (cid:96) ϕ (cid:96) ›› H w(cid:96) ˇˇˇ “ lim t Ñ`8 ˇˇˇ@ e ´ itM (cid:96) ϕ (cid:96) , ` ´ w (cid:96) w ´ j (cid:96) ˘ e ´ itM (cid:96) ϕ (cid:96) D H w(cid:96) ˇˇˇ ď lim t Ñ`8 @ ϕ (cid:96) , e itM (cid:96) p ´ j (cid:96) q e ´ itM (cid:96) ϕ (cid:96) D H w(cid:96) ` lim t Ñ`8 ˇˇˇ@ ϕ (cid:96) , e itM (cid:96) p w ´ w (cid:96) q j (cid:96) w ´ e ´ itM (cid:96) ϕ (cid:96) D H w(cid:96) ˇˇˇ . t Ñ`8 @ ϕ (cid:96) , e itM (cid:96) p ´ j (cid:96) q e ´ itM (cid:96) ϕ (cid:96) D H w(cid:96) ď lim t Ñ`8 @ ϕ (cid:96) , e itM (cid:96) ζ (cid:96) p Q (cid:96) { t q e ´ itM (cid:96) ϕ (cid:96) D H w(cid:96) “ @ ϕ (cid:96) , ζ (cid:96) p V (cid:96) q ϕ (cid:96) D H w(cid:96) “ , while the second term also vanishes by an application of the RAGE theorem. It follows that W ` p M, M (cid:96) , J (cid:96) , I q is isometric on ϕ (cid:96) “ χ p´8 , ´ ε s p V (cid:96) q ϕ (cid:96) , and then a density argument taking into account the first equationin (4.10) implies that W ` p M, M (cid:96) , J (cid:96) , I q is isometric on χ ´ p V (cid:96) q E M (cid:96) p I q H w (cid:96) .A similar proof works for the claims about W ´ p M, M (cid:96) , J (cid:96) , I q and W ˘ p M, M r , J r , I q . The functions η (cid:96) and ζ (cid:96) have to be adapted and the possible negative sign of the variable t has to be taken into account,otherwise the argument can be copied mutatis mutandis. By collecting the results of Theorem 4.6, Remark 4.7, Proposition 4.9, and by using the fact that M (cid:96) and M r have purely absolutely continuous spectrum, one finally obtains a description of the initial setsof the local wave operators W ˘ p M, M , J, I max q : Theorem 4.10.
Let I max : “ σ p M qzt T M Y σ p p M qu and I ‹ : “ σ p M ‹ qz T ‹ ( ‹ “ (cid:96), r ). Then, the local waveoperators W ˘ p M, M , J, I max q : H Ñ H w are partial isometries with initial sets H ˘ : “ χ ¯ p V (cid:96) q E M (cid:96) p I (cid:96) q H w (cid:96) ‘ χ ˘ p V r q E M r p I r q H w r . Remark 4.11.
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