Lifshitz Tails for Generalized Alloy Type Random Schr\"odinger Operators

TL;DR

This paper investigates Lifshitz tails in generalized alloy-type random Schrödinger operators, establishing conditions for their existence and analyzing their behavior at the spectrum's bottom, with applications to random displacement models.

Contribution

It provides a comprehensive analysis of Lifshitz tails for alloy-type potentials with various function forms, including symmetry conditions and their implications.

Findings

Lifshitz tails occur under certain symmetry conditions.

Necessary and sufficient conditions for Lifshitz tails with symmetric potentials.

Application to random displacement models showing Lifshitz singularity.

Abstract

We study Lifshitz tails for random Schr\"odinger 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 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 Lifshitz singularity at the bottom of the spectrum, and also complete the study of continuous Anderson type models undertaken in arXiv : 0804.4079