On quantitative bounds on eigenvalues of a complex perturbation of a Dirac operator

TL;DR

This paper establishes bounds on the eigenvalues of complex-perturbed Dirac operators using complex analysis, extending to Klein-Gordon operators, and addressing challenges posed by unbounded spectra.

Contribution

It introduces a Lieb-Thirring type inequality for complex perturbations of Dirac operators, overcoming the unboundedness issue with complex function theory methods.

Findings

Derived eigenvalue bounds for complex-perturbed Dirac operators.

Extended results to Klein-Gordon operators with complex perturbations.

Addressed the unbounded spectrum challenge in spectral analysis.

Abstract

We prove a Lieb-Thirring type inequality for a complex perturbation of a d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty , -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed operator is not bounded from below in this case, and, to overcome it, we use the methods of complex function theory. The methods of the article also give similar results for complex perturbations of the Klein-Gordon operator.