An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schr\"odinger operators

TL;DR

This paper establishes an eigenvalue estimate for bounded operators and applies it to enhance Lieb-Thirring inequalities for non-selfadjoint Jacobi and Schrödinger operators, advancing spectral analysis techniques.

Contribution

It introduces a new eigenvalue estimate for bounded operators and applies it to improve spectral bounds for non-selfadjoint Jacobi and Schrödinger operators.

Findings

Derived a sum inequality relating eigenvalues and operator distance

Improved existing Lieb-Thirring type inequalities for non-selfadjoint operators

Enhanced spectral bounds for specific classes of operators

Abstract

For bounded linear operators $A,B$ on a Hilbert space $\mathcal{H}$ we show the validity of the estimate $$ \sum_{\lambda \in \sigma_d (B)} \dist(\lambda, \overline{\num}(A))^p \leq \| B-A \|_{\mathcal{S}_p}^p$$ and apply it to recover and improve some Lieb-Thirring type inequalities for non-selfadjoint Jacobi and Schr\"odinger operators.