On the existence of eigenvalues of a one-dimensional Dirac operator

TL;DR

This paper investigates the conditions under which eigenvalues exist within the spectral gap of a one-dimensional Dirac operator with a bounded potential, using variational methods to establish their existence and bounds.

Contribution

It introduces a generalized variational principle to prove the existence and estimate the number of eigenvalues in the spectral gap of the Dirac operator.

Findings

Eigenvalues exist within the spectral gap under certain conditions.

Upper and lower bounds for the eigenvalues are provided.

The number of eigenvalues can be estimated using the developed methods.

Abstract

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.