Vol. 3, No. 4, 2010

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Lifshitz tails for generalized alloy-type random Schrödinger operators

Frédéric Klopp and Shu Nakamura

Vol. 3 (2010), No. 4, 409–426
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.

Keywords
random Schrödinger operators, sign-indefinite potentials, Lifshitz tail
Mathematical Subject Classification 2000
Primary: 35P20, 47B80, 47N55, 81Q10, 82B44
Milestones
Received: 30 March 2009
Revised: 18 February 2010
Accepted: 4 April 2010
Published: 8 September 2010
Authors
Frédéric Klopp
Institut Universitaire de France LAGA, U.M.R. 7539 C.N.R.S
Institut Galilée, Université de Paris-Nord
99 Avenue Jean-Baptiste Clément
93430 Villetaneuse
France
Shu Nakamura
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1 Komaba, Meguro-ku
Tokyo 153-8914
Japan