Analysis & PDE

Poisson statistics for eigenvalues of continuum random Schrödinger operators

Jean-Michel Combes, François Germinet, and Abel Klein

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 3, Number 1 (2010), 49-80.

Dates
Received: 9 July 2009
Accepted: 6 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731040

Digital Object Identifier
doi:10.2140/apde.2010.3.49

Mathematical Reviews number (MathSciNet)
MR2663411

Zentralblatt MATH identifier
1227.82034

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 47B80: Random operators [See also 47H40, 60H25] 60H25: Random operators and equations [See also 47B80]

Keywords
Anderson localization Poisson statistics of eigenvalues Minami estimate level statistics

Citation

Combes, Jean-Michel; Germinet, François; Klein, Abel. Poisson statistics for eigenvalues of continuum random Schrödinger operators. Anal. PDE 3 (2010), no. 1, 49--80. doi:10.2140/apde.2010.3.49. https://projecteuclid.org/euclid.apde/1513731040


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