PE and VL are supported by the Czech Science Foundation (GAČR), Grant No. 17-01706S. PE also acknowledges the support by the European Union within the project CZ.02.1.01/0.0/0.0/16 019/0000778.

The free Dirac operator and associated integral operators {#section_free_Operator}
=========================================================

In this preliminary section we collect some well known facts about the free Dirac operator in $\mathbb{R}^3$ and some associated integral operators that are needed to investigate Dirac operators with singular $\delta$-shell interactions. For that we have to fix some notations first.

Notations
---------

Let $\mathcal{H}$ be a Hilbert space. Then we write $\mathcal{H}^d := \mathcal{H} \otimes \mathbb{C}^d$. For a closable operator $A$ in $\mathcal{H}$ its domain of definition, its range, and its kernel are denoted by ${\mathrm{dom}\,}A$, $\operatorname{ran}A$, and $\ker T$, respectively. The closure of $A$ is $\overline{A}$. Eventually, if $A$ is self-adjoint, then its resolvent set, its spectrum, the point, discrete, and essential spectrum are $\rho(A)$, $\sigma(A)$, $\sigma_\text{p}(A)$, $\sigma_\text{disc}(A)$, and $\sigma_\text{ess}(A)$, respectively.

For a domain $\Omega \subset \mathbb{R}^3$ with a compact $C^2$-smooth boundary $\Sigma := \partial \Omega$ we denote by $L^2(\Omega)$ the standard $L^2$-spaces and $L^2(\Sigma)$ is endowed with the inner product based on the integral with respect to the surface measure $\sigma$. As usual, $H^1(\Omega)$ stands for the Sobolev 