Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

TL;DR

This paper establishes a modified Thouless formula for ergodic Schrödinger operators on the Bethe lattice, connecting the density of states to Lyapunov exponents, and explores the automorphism group and ergodic properties of the lattice.

Contribution

It introduces a modified Thouless formula with a nontrivial remainder term for connectivity greater than one, extending the classical formula from one-dimensional systems.

Findings

The remainder term vanishes at connectivity one, recovering the classical Thouless formula.

The remainder term is nontrivial for connectivity two or more.

The automorphism group of the Bethe lattice influences the ergodic analysis of Schrödinger operators.

Abstract

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $\kappa$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $\kappa \geq 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green's functions along certain paths.