Comptes Rendus
no. 7
On a class of anisotropic asymptotically periodic Hamiltonians
[Hamiltoniens anisotropes asymptotiquement périodiques]
Olivier Rodot1
1 Département de mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France
Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 575-579.

Nous construisons une C * -algèbre ℭ adaptée au traitement des systèmes quantiques anisotropes asymptotiquement périodiques et nous calculons son quotient par l'algèbre des opérateurs compacts. Nous décrivons alors les opérateurs auto-adjoints affiliés à ℭ et leurs spectres essentiels.

We construct a C * -algebra ℭ proper to an anisotropic asymptotically periodic quantum system and we compute its quotient by the algebra of compact operators. We describe then the self-adjoint operators affiliated to ℭ and their essential spectrum.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02301-4
Olivier Rodot 1

1 Département de mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France 
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Olivier Rodot. On a class of anisotropic asymptotically periodic Hamiltonians. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 575-579. doi : 10.1016/S1631-073X(02)02301-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02301-4/

[1] F. Bloch Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys., Volume 52 (1928), pp. 555-600

[2] M. Damak, V. Georgescu, C * -algebras related to the N-body problem and the self-adjoint operators affiliated to them, 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., Volume 63 (1978), pp. 277-301

[4] I.M. Gel'fand Expansion in eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, Volume 73 (1950), pp. 1117-1120 (in Russian)

[5] V. Georgescu, A. Iftimovici, C * -algebras and spectral analysis of quantum systems, in: Proceedings of the OAMP Conference, 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., Volume 31 (1990) no. 9, pp. 2181-2191

[8] E.C. Titchmarsh Eigenfunction Expansions Associated with Second Order Differential Equations, Clarendon Press, Oxford, 1962

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