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Poisson statistics for
eigenvalues of continuum random Schrödinger operators
Jean-Michel Combes, François Germinet and Abel Klein |
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Vol. 3 (2010), No. 1, 49–80
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Abstract |
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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
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Mathematical Subject Classification 2000
Primary: 82B44
Secondary: 47B80, 60H25
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Milestones
Received: 9 July 2009
Accepted: 6 August 2009
Published: 4 March 2010
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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 |
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| http://www.u-cergy.fr/rech/pages/germinet/ | |
| Abel Klein | |
| Department of Mathematics University of California Irvine, CA 92697-3875 United States |
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