\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad
  \sigma_2 := \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad
  \sigma_3 := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ It is easy to see that the Dirac matrices satisfy $$\label{equation_anti_commutation}
  \alpha_j \alpha_k + \alpha_k \alpha_j = 2 \delta_{j k} I_4 \quad \text{and} \quad \alpha_j \beta + \beta \alpha_j = 0, \qquad j, k \in \{ 1, 2, 3\}.$$ For $x = (x_1, x_2, x_3) \in \mathbb{R}^3$ we will often employ the notations $$\alpha \cdot x = \sum_{k=1}^3 \alpha_k x_k \quad \text{and} \quad \alpha \cdot \nabla = \sum_{k=1}^3 \alpha_k \partial_k.$$

Finally, if not stated differently, $\Omega_+ \subset \mathbb{R}^3$ is always a bounded domain with compact $C^2$-smooth boundary $\Sigma$, $\Omega_- = \mathbb{R}^3 \setminus \overline{\Omega_+}$, and $\nu$ denotes the unit normal vector field at $\Sigma$ pointing outwards $\Omega_+$. We will often write $f_\pm := f \upharpoonright \Omega_\pm$ for $f \in L^2(\mathbb{R}^3)$.

The free Dirac operator
-----------------------

We are now prepared to introduce the free Dirac operator, which acts in the Hilbert space $L^2(\mathbb{R}^3)^4$ as $$\label{def_free_Dirac}
      A_0 f := -i c \sum_{j=1}^3 \alpha_j \partial_j f + m c^2 \beta f, 
    \qquad
   {\mathrm{dom}\,}A_0 := H^1(\mathbb{R}^3)^4.
$$ Using the Foldy-Wouthuysen transformation it 