Approximation of Dirac operators with $\boldsymbol{\delta}$-shell potentials in the norm resolvent sense, I. Qualitative results

TL;DR

This paper investigates how regularized operators with sharply peaked potentials approximate Dirac operators with delta-shell interactions in the norm resolvent sense, providing qualitative convergence results for general geometries.

Contribution

It establishes conditions under which regular potentials converge to delta-shell Dirac operators in the norm resolvent sense for complex geometries.

Findings

Convergence of regularized operators to delta-shell operators in norm resolvent sense.

Applicable to general $C^2$-curves and surfaces in $ eal^2$ and $ eal^3$.

Provides qualitative results on approximation of Dirac operators with $oldsymbol{ extdelta}$-shell potentials.

Abstract

In this paper the approximation of Dirac operators with general $\delta$-shell potentials supported on $C^2$-curves in $\mathbb{R}^2$ or $C^2$-surfaces in $\mathbb{R}^3$, which may be bounded or unbounded, is studied. It is shown under suitable conditions on the weight of the $\delta$-interaction that a family of Dirac operators with regular, squeezed potentials converges in the norm resolvent sense to the Dirac operator with the $\delta$-shell interaction.