Lifshitz tails for random diagonal perturbations of Laurent matrices

TL;DR

This paper investigates the Lifshitz tails phenomenon for a class of one-dimensional random operators with long-range interactions, revealing how the spectral edge behavior is influenced by the operator's symbol and the random potential.

Contribution

It extends Lifshitz tail analysis to long-range Laurent matrices with algebraic minima, generalizing existing results to operators with weak off-diagonal decay.

Findings

Lifshitz tails occur at the spectrum's lower edge with an exponent related to the operator's symbol.

The analysis involves generalized Dirichlet-Neumann bracketing for long-range operators.

A generalized Temple's inequality is used to handle degenerate ground states.

Abstract

We study the Integrated Density of States of one-dimensional random operators acting on $\ell^2(\mathbb Z)$ of the form $T + V_\omega$ where $T$ is a Laurent (also called bi-infinite Toeplitz) matrix and $V_\omega$ is an Anderson potential generated by i.i.d. random variables. We assume that the operator $T$ is associated to a bounded, H\"older-continuous symbol $f$, that attains its minimum at a finite number of points. We allow for $f$ to attain its minima algebraically. The resulting operator $T$ is long-range with weak (algebraic) off-diagonal decay. We prove that this operator exhibits Lifshitz tails at the lower edge of the spectrum with an exponent given by the Integrated Density of States of $T$ at the lower spectral edge. The proof relies on generalizations of Dirichlet-Neumann bracketing to the long-range setting and a generalization of Temple's inequality to degenerate ground state energies.