Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory

TL;DR

This paper explores sum rules for Jacobi matrices, providing classifications of spectral measures and proving conjectures related to spectral conditions, with applications to spectral theory.

Contribution

It systematically applies Case's sum rules, including a positive combination, to classify spectral measures and prove Nevai's conjecture for trace class perturbations.

Findings

Complete classification of spectral measures for Hilbert--Schmidt perturbations.

Proof that Szego condition holds if J-J_0 is trace class.

Identification of a positive sum rule combination with special properties.

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 Szego condition holds if J-J_0 is trace class.