Olivier Rodot

Rodot, O.

Hal-Diderot

On a class of anisotropic asymptotically periodic Hamiltonians

HAL Descartes

Rodot, Olivier

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579Équations aux dérivées partielles/Partial Differential Equations(Physique mathématique/Mathematical Physics)On a class of anisotropic asymptotically periodicHamiltoniansOlivier RodotDépartement de mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin,95302 Cergy-Pontoise cedex, FranceReceived 25 January 2002; accepted 4 February 2002Note presented by Jean-Michel Bony.Abstract We construct a C∗-algebra C proper to an anisotropic asymptotically periodic quantumsystem and we compute its quotient by the algebra of compact operators. We describethen the self-adjoint operators affiliated to C and their essential spectrum. To cite thisarticle: O. Rodot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579.  2002 Académiedes sciences/Éditions scientifiques et médicales Elsevier SASHamiltoniens anisotropes asymptotiquement périodiquesRésumé Nous construisons une C∗-algèbre C adaptée au traitement des systèmes quantiquesanisotropes asymptotiquement périodiques et nous calculons son quotient par l’algèbredes opérateurs compacts. Nous décrivons alors les opérateurs auto-adjoints affiliés à C etleurs spectres essentiels. Pour citer cet article : O. Rodot, C. R. Acad. Sci. Paris, Ser. I334 (2002) 575–579.  2002 Académie des sciences/Éditions scientifiques et médicalesElsevier SASVersion française abrégéeConsidérons l’opérateur auto-adjoint H = − + V dans H = L2(R), où V est l’opérateur demultiplication par une fonction asymptotiquement périodique avec des périodes différentes à +∞ et −∞.Les résultats obtenus par Georgescu et Iftimovici dans [6] nous ont suggéré l’étude, par des méthodesnouvelles, de cette classe particulière d’hamiltoniens. L’idée générale consiste à construire une C∗-algèbreC dont le quotient par l’algèbre des opérateurs compacts puisse être calculé, et telle que les opérateurs quel’on veut étudier lui soient affiliés. Dans le cas présent, cette C∗-algèbre C est obtenue grâce à la notion deproduit croisé à partir d’une C∗-algèbre C suggérée par la classe de fonctions V . L’objectif est de préciserla classe la plus large d’hamiltoniens H affiliés à C. Ceci permet l’étude du spectre essentiel ou l’estimationde Mourre d’une manière unifiée. Introduisons à présent la C∗-algèbre commutative C suivante suggéréepar une situation anisotrope :C ={f ∈ Cbu(R) | limn→±∞f (x + na±) existent, ∀x ∈ R},E-mail address:Olivier.Rodot@math.u-cergy.fr (O. Rodot). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-073X(02)02301-4/FLA 575O. Rodot / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579où Cbu(R) = {f : R → C | f est bornée, uniformément continue}, n ∈ Z et (a+, a−) ∈ R∗+ × R∗+. Onmontre que l’on a un plongement canonique : C/C0(R) ⊂ C(R/a+Z)⊕C(R/a−Z). Notons C la C∗-algèbreC  R (produit croisé de C par l’action du groupe additif R). On a alorsC/K(L2(R)) ∼= (C  R)/(C0(R)  R) ∼= (C/C0(R))  Rsi bien que C/K(L2(R)) ⊂ C+ ⊕ C− où C± = C±  R et C± = C(R/a±Z).THÉORÈME 1. – SoitC̃ l’ensemble des opérateursT ∈ B(L2(R)) tels que:(i) il existeT+ ∈ C+ tel que‖χ(Q> r)(T − T+)(∗)‖ → 0 si r → ∞ ;(ii) il existeT− ∈ C− tel que‖χ(Q< −r)(T − T−)(∗)‖ → 0 si r → ∞ ;(iii) ‖(eixP − 1)T (∗)‖ → 0 si x → 0.Alors C̃ est uneC∗-algèbre canoniquement isomorphe àC.THÉORÈME 2. – SoientH un opérateur auto-adjoint dansH = L2(R) et H± un couple d’opérateursauto-adjoints affiliés àC± tels queD(H±) = D(H). AlorsH est affilié àC si∥∥θ+(εQ)(H −H+)∥∥D(H)→H → 0 et ∥∥θ−(εQ)(H −H−)∥∥D(H)→H → 0 si ε → 0.The study of the Schrödinger operator with periodic potential is now a classical subject many timesexplored since the article of Bloch [1] published in 1928. This operator gives a description of the motionof a particle in a crystal. It is well known that the spectrum of this operator has a band structure. Gel’fand[4] and Titchmarsh [8] were among the first to study rigorously the periodic one dimensional Schrödingeroperator H = − + V in H = L2(R) (where the Laplacian , free Hamiltonian, is the quantization ofthe kinetic energy and V is the operator of multiplication by a periodic potential function). More recentlyDavies and Simon have studied in [3] the scattering theory for systems with asymptotic spatial behaviourdifferent on the right and the left. Also Roberts develops in [7] the quantum scattering for impurities inpotentials that tend to a periodic function in one direction and to a constant one in the other. The referenceson this topic are multiple.The new results of Georgescu and Iftimovici in [6] suggested the study to us, with their new methods, ofa class of anisotropic Hamiltonians: the periodic Schrödinger operator with different behaviors at ±∞. Thegeneral idea is to construct a C∗-algebra C, to study its properties and to point out a class H of Hamiltoniansaffiliated to C; this allows one to describe the essential spectrum. This C∗-algebra C is obtained by the notionof crossed product from a C∗-algebra C suggested by the class of functions V . One of the main goals is todetermine the largest class of Hamiltonians H affiliated to C.Let H be a Hilbert space and H a self-adjoint operator in H. Recall that the formula σ(H) = {λ ∈ R |ϕ ∈ C0(R) and ϕ(λ) = 0 ⇒ ϕ(H) = 0} gives a description of the spectrum of H . Here C0(R) = {ϕ : R →C | ϕ is continuous and convergent to zero at infinity}. Recall also that the essential spectrum σess(H) of His the set of λ ∈ σ(H) such that either λ is not isolated from the rest of the spectrum or it is an eigenvalueof infinite multiplicity. We have:σess(H) ={λ ∈ R | ϕ ∈ C0(R) and ϕ(λ) = 0 ⇒ ϕ(H) /∈ K(H)},where K(H) is the C∗-algebra of compact operators in H (closed self-adjoint ideal in the C∗-algebra B(H)of bounded linear operators in H). Let C(H) = B(H)/K(H) be the Calkin algebra (it is also a C∗-algebra).Denote by H → Ĥ the canonical surjection of B(H) onto C(H). Is is easy to see that σess(H) = σ(Ĥ ). IfC is a C∗-subalgebra of B(H), we say that H is affiliated to C if its associated fonctional calculus is in C,i.e., if ∀ϕ ∈ C0(R), ϕ(H) ∈ C. In fact, it suffices to verify for a complex z /∈ σ(H) that (H − z)−1 ∈ C. In[2] and [6] several classes of operators related to interesting physical situations are shown to be affiliated to576To cite this article: O. Rodot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579C∗-algebras C such thatK(H) ⊂ C ⊂ B(H) and C/K(H) ⊂⊕kCk, (1)where Ck are “simpler” C∗-algebras. So, if H is affiliated to a C∗-algebra C which satisfies (1), thenĤ = ⊕k Hk (with Hk affiliated to Ck) satisfies σ(Ĥ ) = ⋃k σ (Hk) with known σ(Hk).Let us consider the C∗-algebra Cbu(R) = {ϕ : R → C | ϕ is bounded, uniformly continuous} and thefollowing commutative C∗-algebra suggested by an anisotropic situation:C ={ϕ ∈ Cbu(R) | limn→±∞ τna±ϕ(x) ≡ l±(x) exist, ∀x ∈ R},where τna±ϕ(x) = ϕ(x + na±), n ∈ Z, and (a+, a−) ∈ R∗+ × R∗+. Then C is a C∗-subalgebra of Cbu(R)which contains C0(R) and is stable by translations. The functions l+ and l− are continuous on R and clearlyperiodic of periods a+ and a−, respectively. We describe now the quotient C∗-algebra C/C0(R).PROPOSITION 1. – C/C0(R) ⊂ C(R/a+Z) ⊕C(R/a−Z).Recall that the self-adjoint operators of L2(R), Q (position observable) and P (momentum observ-able), are defined by (Qf )(x) = xf (x) and (Pf )(x) = −i df/dx . Then ϕ(Q) is the operator of mul-tiplication by the Borel function ϕ in L2(R) and ψ(P) = F∗ψ(Q)F where F : L2(R) → L2(R) withFf (y) = (2π)−1/2 ∫Re−ixyf (x)dx . An elementary computation gives eixPϕ(Q) e−ixP = ϕ(Q + x) andeikQψ(P ) e−ikQ = ψ(P − k). If A, B are subspaces of an algebra D then we denote by A · B the linearsubspace of D generated by the elements of the form AB with A ∈ A and B ∈ B. If D is a C∗-algebra then[[A · B]] is the norm closure of A · B in D. Since C is a C∗-subalgebra of Cbu(R) stable under translations,C is provided with a continuous action of the additive group R. Then the crossed product C  R is welldefined and denoted by C. It is shown in [6] that C is isomorphic to the (norm) closed linear subspace ofB(L2(R)) generated by operators of the form ϕ(Q)ψ(P) with ϕ ∈ C and ψ ∈ C0(R). Below we denoteC0(R∗) the set of operators ψ(P) with ψ ∈ C0(R). We thus may write C = [[C · C0(R∗)]]. One also haveK(L2(R)) = [[C0(R) ·C0(R∗)]] so C0(R)  R is isomorphic to K(L2(R)). ThenC/K(L2(R)) ∼= (C  R)/(C0(R)  R) ∼= (C/C0(R))  Rso that C/K(L2(R)) ⊂ C+ ⊕ C− where C± = C±  R and C± = C(R/a±Z). Recall again that C± =[[C± ·C0(R∗)]].Now we give a new characterization of C. If a symbol like T (∗) is used then the relation must hold bothfor the operator T and for its adjoint T ∗. We denote χ(A > r) the spectral projection of a self-adjointoperator A associated to the interval (r,∞). The symbol χ(A < r) has a similar meaning.THEOREM 2. – C coincides with the set of the operatorsT ∈ B(L2(R)) such that(i) there is someT+ ∈ C+ such that‖χ(Q> r)(T − T+)(∗)‖ → 0 if r → ∞;(ii) there is someT− ∈ C− such that‖χ(Q< −r)(T − T−)(∗)‖ → 0 if r → ∞;(iii) ‖(eixP − 1)T (∗)‖ → 0 if x → 0.Proof. –Let C̃ be the set of operators T satisfying (i)–(iii); we first show that C ⊂ C̃. Since C is isomorphicto the (norm) closed linear subspace of B(L2(R)) generated by operators of the form ϕ(Q)ψ(P) with ϕ ∈ Cand ψ ∈ C0(R∗), our task reduces to show that these operators belong to C̃. So let T = ϕ(Q)ψ(P) and setT± = l±(Q)ψ(P ). It suffices to show (i) and (ii) for r = na+.∥∥χ(Q> na+)(T − T+)∥∥ = ∥∥e−ina+P χ(Q> 0) eina+P (ϕ(Q)ψ(P) − l+(Q)ψ(P ))∥∥= ∥∥χ(Q> 0) eina+P (ϕ(Q)ψ(P) − l+(Q)ψ(P )) e−ina+P∥∥.But eina+P l+(Q)ψ(P ) e−ina+P l+(Q + na+)ψ(P ) = l+(Q)ψ(P ) since l+ is periodic of period a+.Similarly eina+P ϕ(Q)ψ(P) e−ina+P = ϕ(Q+ na+)ψ(P ).577O. Rodot / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579∥∥χ(Q> na+)(T − T+)∥∥ = ∥∥χ(Q> 0)(ϕ(Q + na+)− l+(Q))ψ(P)∥∥ supx>0∣∣ϕ(x + na+)− l+(x)∣∣‖ψ‖∞ → 0 as n → ∞.The assertion (iii) is a consequence of ψ ∈ C0(R∗) (see[2] and [5]).Reciprocally, let T ∈ C̃. Let θ+ ∈ C∞(R), 0  θ+  1, such that θ+(x) = 0 if x < 1 and θ+(x) = 1 ifx > 2. Set θ−(x) = θ+(−x), θ0 = 1 − θ− − θ+, θε± = θ±(εQ) and θε0 = θ0(εQ). (i) and (ii) are equivalent to‖θε±(T −T±)(∗)‖ → 0 if ε → 0 respectively. Since T = θε0T + θε+T+ + θε−T− + θε+(T −T+)+ θε−(T −T−),the last two terms tend to zero in norm when ε → 0. Also, θε± ∈ C yelds θε±T± ∈ C. Finally we haveto prove that θε0T ∈ K(L2(R)) ⊂ C. By a characterization of K(L2(R)) given in [6], θε0T ∈ K(L2(R))iff ‖(eixP − 1)θε0T ‖ → 0 as x → 0 and ‖(eikQ − 1)θε0T ‖ → 0 as k → 0. But ‖(eixP − 1)θε0T ‖ ‖[eixP , θε0 ]‖‖T ‖ + ‖θε0‖‖(eixP − 1)T ‖. Since ‖(eixP − 1)T ‖ → 0 by (iii), and∥∥[eixP , θε0 ]∥∥ = ∥∥eixP θε0 e−ixP − θε0∥∥ = ∥∥θε0 (· + x)− θε0∥∥∞ → 0 as x → 0.We get ‖(eikQ−1)θε0T ‖  ‖(eikQ−1)θε0‖‖T ‖ = ‖(eik·−1)θε0‖∞‖T ‖ → 0 if k → 0, thus θε0T ∈ K(L2(R)),which finishes the proof of the theorem. ✷We shall point out now an affiliation criterion to C for a self-adjoint operator. We first introduce someuseful definitions (see[5]).DEFINITION 3. – An operator T ∈ B(L2(R)) will be called semi-compactif for all θ ∈ C0(R) theoperators θ(Q)T and T θ(Q) are compact. The self-adjoint operators affiliated to SK(L2(R)) are calledlocally compact.Remark that the set SK(L2(R)) of semi-compact operators is a C∗-subalgebra of B(L2(R)) and thatC± ⊂ HK(L2(R)) ⊂ SK(L2(R)) whereHK(L2(R)) = {T ∈ B(L2(R)) | limr→∞(∥∥χ(|P | > r)T ∥∥ + ∥∥T χ(|P | > r)∥∥) = 0}.DEFINITION 4. – We say that T ∈ B(L2(R)) has limit T± at Q = ±∞ (and set limQ→±∞T = T±) iflimε→0{‖θ±(εQ) (T − T±)‖ + ‖(T − T±)θ±(εQ)‖} = 0.PROPOSITION 5. – C = {T ∈ SK(L2(R)) | limQ→±∞ T exist and belong toC±}.COROLLARY 6. – The following assertions are equivalent:(i) H self-adjoint operator affiliated toC.(ii) H is locally compact and∃z ∈ C \ R such thatlimQ→±∞(H − z)−1 exist and are inC±.The next proposition gives a method for checking local compactness.PROPOSITION 7. – Assume thatK is a Banach space continuously embedded inH and such thatθ(Q) ∈ K(K,H) for eachθ ∈ C∞c (R). If H is a self-adjoint operator inH and (H + i)−1H ⊂ K, thenH is locally compact. In particular, if the domain ofH is included in the form domain of a locally compactoperator, thenH is locally compact.Our second important result states as follows:THEOREM 8. – LetH be a self-adjoint operator inH = L2(R) andH± a pair of self-adjoint operatorsaffiliated toC± respectively such thatD(H±) = D(H). Assume that∥∥θ+(εQ)(H −H+)∥∥D(H)→H → 0 and ∥∥θ−(εQ)(H −H−)∥∥D(H)→H → 0 if ε → 0.ThenH is affiliated toC. In particularσess(H) = σ(H+)∪ σ(H−).578Pour citer cet article : O. Rodot, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 575–579Proof. –Since H± are affiliated to C± ⊂ SK(H), H± are locally compact. H is locally compact becauseof D(H±) = D(H) and of Proposition 7. So, by Corollary 6, it suffices to prove that limQ→±∞(H −z)−1 =(H± − z)−1. We denote R(z) = (H − z)−1 and R±(z) = (H± − z)−1. Then∥∥θ±(εQ)(R −R±)∥∥ = ∥∥θ±(εQ)R±(H −H±)R∥∥∥∥[θ±(εQ),R±]∥∥ · ∥∥(H −H±)R∥∥ + ‖R±‖ · ∥∥θ±(εQ)(H −H±)R∥∥.But ‖θ±(εQ) (H −H±)R ‖ → 0 when ε → 0 because of the hypothesis. On the other hand, ‖[θ±(εQ),R±]‖ (ε/√2π) · ‖θ̂ ′±‖L1(R) ·∥∥R′±∥∥L∞(R) , thus ‖θ±(εQ) (R −R±)‖ → 0 when ε → 0. ✷We give now an example. Let h : R → R be a continuous function such that c−1|x|s  |h(x)|  c|x|s forsome real s > 0, a constant c > 0 and large x . Set H0 = h(P ), its domain being the Sobolev space Hs(R).Let V be a real function such that V (Q) ∈ B(Hs ,H) with H0-bound less than 1 and assume that there area±-periodic functions V± such that ‖θ±(εQ) (V − V±)‖Hs→H → 0 if ε → 0. Then H = H0 + V (Q) isaffiliated to C and H± = H0 + V±(Q), soσess(H) = σ(H+)∪ σ(H−).Remark. – If E is a Hilbert space, then one can replace above C by C⊗K(L2(E)). Thus our results coverthe case of one-dimensional Dirac operators for example (then E is finite dimensional).Acknowledgments. I express my gratitude to V. Georgescu, A. Iftimovici and S. Golénia for their helpful advice.References[1] F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52 (1928) 555–600.[2] M. Damak, V. Georgescu, C∗-algebras related to the N -body problem and the self-adjoint operators affiliated tothem, available as preprint 99-482 at http://www.ma.utexas.edu/mp_arc/.[3] E.B. Davies, B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Comm.Math. Phys. 63 (1978) 277–301.[4] I.M. Gel’fand, Expansion in eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR 73(1950) 1117–1120 (in Russian).[5] V. Georgescu, A. Iftimovici, C∗-algebras and spectral analysis of quantum systems, in: Proceedings of the OAMPConference, Constanta, 2001, to appear.[6] V. Georgescu, A. Iftimovici, C∗-algebras of energy observables, Preprint 9/2001, Université de Cergy-Pontoise,pp. 1–120, also available at http://www.ma.utexas.edu/mp_arc.[7] T.M. Roberts, Scattering for step-periodic potentials in one dimension, J. Math. Phys. 31 (9) (1990) 2181–2191.[8] E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, Clarendon Press,Oxford, 1962.579

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/S1631-073X(02)02301-4.pdf

On a class of anisotropic asymptotically periodic Hamiltonians

Abstract

Similar works

Full text

Available Versions

Hal-Diderot

HAL Descartes

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique