Spectral transition for Dirac operators with electrostatic $\delta$-shell potentials supported on the straight line

TL;DR

This paper investigates how the spectrum of a 2D Dirac operator with a line-supported electrostatic delta potential changes at critical interaction strengths, revealing a sudden collapse of the continuous spectrum to a point.

Contribution

It provides a detailed analysis of spectral transitions in Dirac operators with delta-shell interactions, highlighting the critical strengths where the spectrum collapses.

Findings

Spectral transition occurs at critical strengths η=±2.

Continuous spectrum collapses to a single point at these critical values.

The study characterizes the spectral behavior of Dirac operators with delta potentials.

Abstract

In this note the two dimensional Dirac operator $A_\eta$ with an electrostatic $\delta$-shell interaction of strength $\eta\in\mathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $\eta=\pm 2$ the continuous spectrum of $A_\eta$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.