Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions

TL;DR

This paper investigates how critical shell interactions influence the essential spectrum of 3D Dirac operators, revealing a new spectral interval determined by surface curvature and coupling constants, a novel phenomenon in this context.

Contribution

It introduces the first analysis of a new essential spectrum interval caused by critical shell interactions in three-dimensional Dirac operators, linking spectral properties to surface geometry.

Findings

Critical interactions create a new essential spectrum interval.

The interval's position and size depend on coupling constants and surface curvatures.

This phenomenon is unique compared to lower-dimensional or special-geometry cases.

Abstract

We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. The position and the length of the interval are explicitly controlled by the coupling constants and the principal curvatures of the surface. This effect is completely new compared to lower dimensional critical situations or special geometries considered up to now, in which only a single new point in the essential spectrum was observed.