Monotonicity of the set of zeros of the Lyapunov exponent with respect to shift embeddings

TL;DR

This paper investigates the zeros of the Lyapunov exponent for discrete Schrödinger operators with shift-based potentials, proving their monotonic behavior under shift embeddings and introducing a related monotonic function.

Contribution

It establishes the monotonicity of the zero set of the Lyapunov exponent and a related function with respect to shift embeddings, advancing understanding of spectral properties.

Findings

Monotonicity of the zero set of the Lyapunov exponent under shift embeddings

Introduction of the function ${ m extbf J}(A, extbf u)$ and proof of its monotonicity

Insights into spectral characteristics of Schrödinger operators with shift-based potentials

Abstract

We consider the discrete Schr\"odinger operators with potentials whose values are read along the orbits of a shift of finite type. We study a certain subset of the collection of energies at which the Lyapunov exponent is zero and prove monotonicity of this set with respect to the shift embeddings. Then we introduce a certain function ${\mathcal J}(A,\mu)$ determined by the position of these zeros and prove monotonicity of ${\mathcal J}(A,\mu)$ with respect to embeddings.