Analysis & PDE

  • Anal. PDE
  • Volume 3, Number 4 (2010), 409-426.

Lifshitz tails for generalized alloy-type random Schrödinger operators

Frédéric Klopp and Shu Nakamura

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Abstract

We study Lifshitz tails for random Schrödinger operators where the random potential is alloy-type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have a Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models.

Article information

Source
Anal. PDE, Volume 3, Number 4 (2010), 409-426.

Dates
Received: 30 March 2009
Revised: 18 February 2010
Accepted: 4 April 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731095

Digital Object Identifier
doi:10.2140/apde.2010.3.409

Mathematical Reviews number (MathSciNet)
MR2718259

Zentralblatt MATH identifier
1226.35058

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47B80: Random operators [See also 47H40, 60H25] 47N55 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
random Schrödinger operators sign-indefinite potentials Lifshitz tail

Citation

Klopp, Frédéric; Nakamura, Shu. Lifshitz tails for generalized alloy-type random Schrödinger operators. Anal. PDE 3 (2010), no. 4, 409--426. doi:10.2140/apde.2010.3.409. https://projecteuclid.org/euclid.apde/1513731095


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