Sum Rules for Jacobi Matrices and Divergent Lieb-Thirring Sums

TL;DR

This paper develops new sum rules for Jacobi matrices linking spectral measures and eigenvalues, leading to results on the divergence of Lieb-Thirring sums and Szego integrals, extending prior theoretical work.

Contribution

It introduces novel sum rules for Jacobi matrices that relate spectral measures and eigenvalues, advancing understanding of spectral divergence phenomena.

Findings

Lieb-Thirring sums of eigenvalues can diverge under certain conditions.

Szego-type integrals of the absolutely continuous spectrum are shown to diverge.

New sum rules connect spectral measure properties with eigenvalue distributions.

Abstract

Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to obtain various results on divergence of certain Lieb-Thirring sums of eigenvalues and Szego-type integrals of the a.c. part of the spectral measure.