Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

TL;DR

This paper establishes eigenvalue bounds for non-selfadjoint Dirac operators on the real line, showing eigenvalues lie in specific regions under certain potential norms, with implications for resonance and embedded eigenvalue estimates.

Contribution

It provides new eigenvalue estimates for non-Hermitian Dirac operators, extending known results to potentials with slower decay and analyzing the nonrelativistic limit.

Findings

Eigenvalues lie in two disjoint disks in the complex plane.

Massless Dirac operators have no nonreal eigenvalues under the given conditions.

Bounds on resonances and embedded eigenvalues are derived for specific potentials.

Abstract

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schr\"odinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.