Spectral measures of Jacobi operators with random potentials

Abstract

Let $H_ω$ be a self-adjoint Jacobi operator with a potential sequence $\{ω(n)\}_n$ of independently distributed random variables with continuous probability distributions and let $μ_ϕ^ω$ be the corresponding spectral measure generated by $H_ω$ and the vector $ϕ$. We consider sets $A(ω)$ which depend on $ω$ in a particular way and prove that $μ_ϕ^ω(A(ω))=0$ for almost every $ω$. 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.