Vol. 3, No. 1, 2010

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Poisson statistics for eigenvalues of continuum random Schrödinger operators

Jean-Michel Combes, François Germinet and Abel Klein

Vol. 3 (2010), No. 1, 49–80
Abstract

We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. In addition, we prove that in this localization region the eigenvalues are simple.

These results rely on a Minami estimate for continuum Anderson Hamiltonians. We also give a simple, transparent proof of Minami’s estimate for the (discrete) Anderson model.

Keywords
Anderson localization, Poisson statistics of eigenvalues, Minami estimate, level statistics
Mathematical Subject Classification 2000
Primary: 82B44
Secondary: 47B80, 60H25
Milestones
Received: 9 July 2009
Accepted: 6 August 2009
Published: 4 March 2010
Authors
Jean-Michel Combes
Département de Mathématiques
Université du Sud: Toulon-Var
83130 La Garde
France
Centre de Physique Théorique
CNRS Luminy
Case 907
13288 Marseille
France
François Germinet
Département de Mathématiques
Université de Cergy-Pontoise
95000 Cergy-Pontoise
France
http://www.u-cergy.fr/rech/pages/germinet/
Abel Klein
Department of Mathematics
University of California
Irvine, CA 92697-3875
United States