The Dirac operator is a first order partial differential operator acting in
the
space of vector valued functions $L^2(\mathbb{R}^3; \mathbb{C}^4)$.
It is the relativistic counterpart of the Schr\"odinger operator and
hence, it appears in many applications in quantum mechanics.
In this talk I will discuss the Dirac operator with an electrostatic
$\delta$-shell interaction which is formally given by $A_\eta := A_0 + \eta
\delta_\Sigma I_4$,
where $A_0$ is the free Dirac operator, $I_4$ is the $4 \times 4$ identity
matrix, $\eta \in \mathbb{R}$
and $\Sigma$ is the boundary of a smooth domain.
After establishing the self-adjointness of $A_\eta$ it turns out that some
of the
spectral properties of $A_\eta$ are of a different nature, if $\eta = 2 c$,
where $c$ denotes the speed of light,
or $\eta \neq 2 c$.
In the latter case functions in $\mathrm{dom}\, A_\eta$ are $H^1$-smooth in
$\mathbb{R}^3 \setminus \Sigma$,
the discrete spectrum is finite and the nonrelativistic limit of $A_\eta$,
as the speed of light $c$ tends to $\infty$, can be computed.
In the case $\eta = 2 c$ these properties either differ from those for $\eta
\neq 2 c$ or they are still unknown.
This talk is based on joint works with J. Behrndt, P. Exner, and V.
Lotoreichik.