# The relativistic spherical $\delta$-shell interaction in $\mathbb{R}^3$: spectrum and approximation

## TL;DR

This paper studies the spectral properties of the Dirac operator with a spherical delta-shell interaction, characterizes eigenstates through inequalities, and shows the equivalence of different domain definitions, also analyzing approximation by short-range potentials.

## Contribution

It provides a detailed spectral analysis of spherical delta-shell interactions for Dirac operators, clarifies domain equivalences, and improves approximation results by short-range potentials.

## Key findings

- Eigenstates characterized by sharp inequalities.
- Domains of different definitions are shown to coincide.
- Spectral relation between shell interaction and short-range potential approximation.

## Abstract

This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich, Exner and \v{S}eba [Dirac operators with a spherically symmetric $\delta$-shell interaction, J. Math. Phys. 30.12 (1989), 2875-2882] and by Arrizabalaga, Mas and Vega [Shell interactions for Dirac operators, J. Math. Pures et Appl. 102.4 (2014), 617-639] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.