On the stability and spectral properties of the two-dimensional Brown-Ravenhall operator with a short-range potential

TL;DR

This paper analyzes the stability and spectral characteristics of a two-dimensional Brown-Ravenhall operator with a short-range $K_0$-potential, revealing conditions for boundedness and spectral properties including eigenvalues and essential spectrum.

Contribution

It introduces a modified 2D Brown-Ravenhall operator with $K_0$-potential and studies its stability and spectral features, including boundedness and eigenvalue existence.

Findings

The operator is bounded from below for coupling constants below a critical value.

The operator remains bounded below even above the critical coupling.

The spectrum's essential part and eigenvalues are characterized, including embedded eigenvalues.

Abstract

The Brown-Ravenhall operator was initially proposed as an alternative to describe the fermion-fermion interaction via Coulomb potential and subject to relativity. This operator is defined in terms of the associated Dirac operator and the projection onto the positive spectral subspace of the free Dirac operator. In this paper, we propose to analyze a modified version of the Brown-Ravenhall operator in two-dimensions. More specifically, we consider the Brown-Ravenhall operator with a short-range attractive potential given by a Bessel-Macdonald function (also known as $K_0$-potential) using the Foldy-Wouthuysen unitary transformation. Initially, we prove that the two-dimensional Brown-Ravenhall operator with $K_0$-potential is bounded from below when the coupling constant is below a specified critical value (a property also referred to as stability). A major feature of this model is the fact that it does not cease to be bounded below even if the coupling constant is above the specified critical value. We also investigate the nature of the spectrum of this operator, in particular the location of the essential spectrum, and the existence of eigenvalues, which are either isolated from the essential spectrum or embedded in it.