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Nonperturbative Techniques in the Investigation of the Spectral Properties of Many-Channel Systems

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Partial Differential Equations and Spectral Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 126))

Abstract

We show that the C*-algebras generated by the “virtual” hamiltonians of a quantum system (with a qualitatively specified interaction) has in many cases an interesting and nontrivial structure. In particular, the quotient of this algebra with respect to the ideal of compact operators can be explicitly computed. This allows one to determine in a unified way the essential spectrum and to prove the Mourre estimate for large classes of hamiltonians, including: N-Body systems, stratified media, particles subject to Klaus type interactions (widely separated bumps) and other classes of (phase-space) anisotropic hamiltonians. The results presented here are based on a joint work with V. Georgescu.

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References

  1. W. Amrein, A. Boutet de Monvel and V. Georgescu, C o -groups, commutator methods and spectral theory of N-body hamiltonians, Birkhauser Verlag, 1996.

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  2. M. Damak and 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. V. Georgescu, A. Iftimovici C*-algebras of energy observables I. The essentialspectrum (in preparation)

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  4. M. Klaus, On -d 2 /dx 2 +V where V has infinitely many “bumps”, Ann. Inst.H. Poincaré Sect. A (N.S.) 38 (1983), no. 1, 7–13.

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Author information

Authors and Affiliations

  1. Département de Mathématiques de l’Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, F 95302, Cergy Pontoise Cedex, France

    Andrei Iftimovici

Authors
  1. Andrei Iftimovici
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, TU Clausthal, Erzstr. 1, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  2. Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, Potsdam, 14469, Germany

    Bert-Wolfgang Schulze

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© 2001 Springer Basel AG

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Iftimovici, A. (2001). Nonperturbative Techniques in the Investigation of the Spectral Properties of Many-Channel Systems. In: Demuth, M., Schulze, BW. (eds) Partial Differential Equations and Spectral Theory. Operator Theory: Advances and Applications, vol 126. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8231-6_18

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  • DOI: https://doi.org/10.1007/978-3-0348-8231-6_18

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9483-8

  • Online ISBN: 978-3-0348-8231-6

  • eBook Packages: Springer Book Archive

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