An Inverse Spectral Problem for Non-Self-Adjoint Jacobi Matrices

Abstract

We consider the class of bounded symmetric Jacobi matrices $J$ with positive off-diagonal elements and complex diagonal elements. With each matrix $J$ from this class, we associate the spectral data, which consists of a pair $(\nu ,\psi )$. Here $\nu $ is the spectral measure of $|J|=\sqrt {J^{*}J}$ and $\psi $ is a phase function on the real line satisfying $|\psi |\leq 1$ almost everywhere with respect to the measure $\nu $. Our main result is that the map from $J$ to the pair $(\nu ,\psi )$ is a bijection between our class of Jacobi matrices and the set of all spectral data.