Spectral measures of Jacobi operators with random potentials

TL;DR

This paper investigates the spectral properties of Jacobi operators with random potentials, demonstrating that certain spectral measures vanish almost surely and exploring the relationships between spectral measures and eigenvalues of these operators.

Contribution

It establishes that specific spectral measures are almost surely zero for random Jacobi operators and analyzes the connections between spectral measures and eigenvalues of the matrices and their submatrices.

Findings

Spectral measures of random Jacobi operators vanish on certain sets almost surely.

The paper shows equivalence relations between spectral measures of these operators.

It studies the relationship between eigenvalues of the operators and their submatrices.

Abstract

Let $H_\omega$ be a self-adjoint Jacobi operator with a potential sequence $\{\omega(n)\}_n$ of independently distributed random variables with continuous probability distributions and let $\mu_\phi^\omega$ be the corresponding spectral measure generated by $H_\omega$ and the vector $\phi$. We consider sets $A(\omega)$ which depend on $\omega$ in a particular way and prove that $\mu_\phi^\omega(A(\omega))=0$ for almost every $\omega$. This is applied to show equivalence relations between spectral measures for random Jacobi matrices and to study the interplay of the eigenvalues of these matrices and their submatrices.