On two-dimensional Dirac operators with critical delta-shell interactions

TL;DR

This paper investigates the spectral properties of two-dimensional Dirac operators with critical delta-shell interactions on lines and circles, revealing distinct behaviors at critical interaction strengths and proposing conjectures for more general geometries.

Contribution

It provides a detailed analysis of the spectral nature at critical interaction strengths for Dirac operators on specific geometries, highlighting differences between line and circle cases.

Findings

Critical points can be eigenvalues of infinite multiplicity or accumulation points.

The nature of the spectral point differs between line and circle geometries.

Findings suggest new conjectures for generic smooth curves.

Abstract

We study two-dimensional Dirac operators with singular interactions of electrostatic and Lorentzscalar type, supported either on a straight line or a circle. For certain critical values of the interaction strengths, the essential spectrum of such operators comprises an isolated point lying within the mass gap. We clarify the nature of this point in both geometries. For the straight line model, this point is known to be an eigenvalue of infinite multiplicity, and we provide a detailed analysis of the corresponding eigenfunctions. By contrast, in the case of a circle, we show that the said point is not itself an eigenvalue, but rather an accumulation point of a double sequence of simple eigenvalues. In view of the high degree of symmetry of the configurations under analysis, this behavior is unexpected and our findings lead us to formulate some conjectures concerning critical singular interactions supported on generic smooth curves.