Sum rules for Jacobi matrices and their applications to spectral theory

Abstract

We discuss the proof of and systematic application of Case’s sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices $J$ for which $J-J_0$ is Hilbert-Schmidt, and a proof of Nevai’s conjecture that the Szegő condition holds if $J-J_0$ is trace class.