On the approximation of the $\delta$-shell interaction for the 3-D Dirac operator

TL;DR

This paper studies how to approximate a 3D Dirac operator with delta-shell interactions using local potentials, showing convergence in the strong resolvent sense without smallness assumptions, but with nonlinear coupling dependence.

Contribution

It demonstrates the strong resolvent convergence of general local interactions to the Dirac operator with delta-shell coupling, revealing nonlinear coupling dependence.

Findings

Convergence in the strong resolvent sense without smallness assumptions

Coupling constant depends nonlinearly on the potential V

Approximation of delta-shell interactions with local potentials

Abstract

We consider the three-dimensional Dirac operator coupled with a combination of electrostatic and Lorentz scalar $\delta$-shell interactions. We approximate this operator with general local interactions $V$. Without any hypotheses of smallness on the potential $V$, converges in the strong resolvent sense to the Dirac Hamiltonian coupled with a $\delta$-shell potential supported on $\S$, a bounded smooth surface. However, the coupling constant depends nonlinearly on the potential $V.$