Critical Lieb-Thirring Bounds in Gaps and the Generalized Nevai Conjecture for Finite Gap Jacobi Matrices

TL;DR

This paper establishes Lieb-Thirring type bounds in spectral gaps for finite gap Jacobi matrices and continuum Schrödinger operators, linking $L^1$ perturbation conditions to spectral properties and Szegő conditions.

Contribution

It introduces a new form of the Birman--Schwinger bound in gaps and proves a generalized Nevai conjecture relating $L^1$ perturbations to spectral conditions.

Findings

Derived bounds for eigenvalues in spectral gaps.

Extended Lieb-Thirring inequalities to finite gap Jacobi matrices.

Connected $L^1$ perturbation norms to Szegő conditions.

Abstract

We prove bounds of the form $\sum_{e\in I\cap\sigma_\di (H)} \dist (e,\sigma_\e (H))^{1/2} \leq L^1$-norm of a perturbation, where $I$ is a gap. Included are gaps in continuum one-dimensional periodic Schr\"odinger operators and finite gap Jacobi matrices where we get a generalized Nevai conjecture about an $L^1$ condition implying a Szeg\H{o} condition. One key is a general new form of the Birman--Schwinger bound in gaps.