Density of States and Thouless Formula for Random Unitary Band Matrices

TL;DR

This paper investigates the density of states for specific random unitary band matrices, establishing a Thouless formula linking it to Lyapunov exponents, and analyzes the measure's support and analyticity conditions.

Contribution

It introduces a Thouless formula for these matrices, connecting spectral properties with dynamical stability, and characterizes the measure's support and conditions for analytic density.

Findings

Derived a Thouless formula relating density of states to Lyapunov exponent.

Determined the support of the density of states measure.

Identified conditions for the measure to have an analytic density.

Abstract

We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can also be considered as a unitary version of the Anderson model. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.