Perturbation of eigenvalues of the Klein-Gordon operators

TL;DR

This paper develops spectral inclusion theorems and bounds for Klein-Gordon operators, providing insights into eigenvalue behavior under potential changes and extending to Sturm-Liouville problems with indefinite weights.

Contribution

It introduces a general framework for spectral analysis of Klein-Gordon operators and applies it to eigenvalue bounds and other operator classes.

Findings

Established inclusion theorems for spectra and essential spectra.

Derived two-sided bounds for isolated eigenvalues.

Applied results to Sturm-Liouville problems with indefinite weights.

Abstract

We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study operators of the form $JG$, where $J$, $G$ are selfadjoint operators on a Hilbert space, $J = J^* = J^{-1}$ and $G$ is positive definite and then we apply these results to obtain bounds of the Klein-Gordon eigenvalues under the change of the electrostatic potential. The developed general theory allows applications to some other instances, as e.g. the Sturm-Liouville problems with indefinite weight.