Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices

TL;DR

This paper establishes bounds on eigenvalues in spectral gaps of Schrödinger operators and Jacobi matrices, extending classical results to more general perturbations and indefinite cases.

Contribution

It introduces a general eigenvalue bound theorem for selfadjoint operators with perturbations of indefinite sign, applied to Schrödinger and Jacobi operators.

Findings

Derived a $( ext{perturbation})^{d/2}$ eigenvalue bound for Schrödinger operators.

Established a Lieb-Thirring type bound for Jacobi matrices.

Extended eigenvalue bounds to operators with indefinite perturbations.

Abstract

We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations of suitable periodic Schrodinger operators and a (not quite)Lieb-Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.