Lieb-Thirring Inequalities for Jacobi Matrices

TL;DR

This paper establishes Lieb-Thirring inequalities for Jacobi matrices, providing bounds on the sum of spectral moments in terms of matrix coefficients, with extensions to higher moments and dimensions.

Contribution

It introduces new Lieb-Thirring inequalities for Jacobi matrices, including bounds on higher moments and results in higher-dimensional settings.

Findings

Proved spectral sum bounds for Jacobi matrices in terms of their coefficients.

Extended inequalities to higher moments of the spectrum.

Derived related results applicable in higher-dimensional contexts.

Abstract

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher dimension.