# Dirac operators and shell interactions: a survey

## TL;DR

This survey reviews recent developments in Dirac operators with delta-shell interactions, focusing on self-adjointness, approximation by squeezing potentials, spectral properties, and asymptotic eigenvalue behavior.

## Contribution

It compiles and discusses recent advances in understanding the mathematical properties and spectral analysis of Dirac operators with shell interactions.

## Key findings

- Self-adjointness conditions for Dirac operators with delta-shells
- Approximation of these operators via squeezing potentials
- Asymptotic behavior of eigenvalues in spectral gaps

## Abstract

In this survey we gather recent results on Dirac operators coupled with $\delta$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials ? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled with a Lorentz-scalar interaction.