Spectral measures of Jacobi operators with random potentials

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 ${\mu }_{\varphi }^{\omega }$ be the corresponding spectral measure generated by $H_{\omega }$ and the vector $\varphi$. We consider sets $A(\omega )$ which depend on $\omega$ in a particular way and prove that ${\mu }_{\varphi }^{\omega }(A(\omega ))=0$ 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.

15 pages; references updated; typos corrected