Low energy properties of the random displacement model

TL;DR

This paper investigates the low-energy behavior of the random displacement model, a Schrödinger operator with randomly deformed lattices, revealing unique spectral properties and asymptotics in different dimensions.

Contribution

It characterizes all minimizing configurations of the model and uncovers unusual low-energy asymptotics, especially in one dimension with Bernoulli-distributed displacements.

Findings

Unique minimizing configurations in higher dimensions

Infinitely many minimizers in one dimension

Logarithmic singularity in the density of states for Bernoulli displacements

Abstract

We study low-energy properties of the random displacement model, a random Schr\"odinger operator describing an electron in a randomly deformed lattice. All periodic displacement configurations which minimize the bottom of the spectrum are characterized. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states of the one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it has a $1/\log^2$-singularity at the bottom of the spectrum. In particular, it is not H\"older-continuous.