# Approximation of one-dimensional relativistic point interactions by regular potentials revised

## TL;DR

This paper demonstrates that one-dimensional relativistic point interactions can be approximated by scaled regular potentials, with the approximation's limit being independent of the specific potential shape, in the norm resolvent sense.

## Contribution

It provides a rigorous method to approximate relativistic point interactions using scaled regular potentials, extending understanding of their stability and dependence.

## Key findings

- Approximation in the norm resolvent sense is possible.
- The limit is independent of the specific shape of the regular potential.
- The approach applies to a broad class of point interactions.

## Abstract

We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form $1/\varepsilon~h(x/\varepsilon)\otimes B$, where $B$ is a suitable $2\times 2$ matrix. Moreover, we prove that the limit does not depend on the particular choice of $h$ as long as it integrates to a constant value.