\textup{d} y, \quad
  f \in L^2(\mathbb{R}^3)^4,~x \in \Sigma.$$ It follows from , , and that $\Phi_\lambda^* f \in H^{1/2}(\Sigma)^4$ for any $f \in L^2(\mathbb{R}^3)^4$.

Definition and self-adjointness of $A_{\eta, \tau}$ {#section_self_adjoint}
===================================================

This section is devoted to the rigorous mathematical definition of the operator $A_{\eta, \tau}$ and the proof of its self-adjointness. In the following we will often make use of the orthogonal decomposition $L^2(\mathbb{R}^3)^4 = L^2(\Omega_+)^4 \oplus L^2(\Omega_-)^4$ and we write for $f \in L^2(\mathbb{R}^3)^4$, in this sense, $f = f_+ \oplus f_-$ with $f_\pm := f \upharpoonright \Omega_\pm$.

As explained in the introduction, see , the $\delta$-shell interaction is modeled by a coupling condition which has to be satisfied by functions in the operator domain. We define for $\eta, \tau \in \mathbb{R}$ the operator $A_{\eta, \tau}$ by $$\label{def_A_eta}
    \begin{split}
      A_{\eta, \tau} f 
        &:= (-i c \alpha \cdot \nabla + m c^2 \beta) f_+ \oplus (-i c \alpha \cdot \nabla + m c^2 \beta) f_-, \\
      {\mathrm{dom}\,}A_{\eta, \tau} 
          &:= \big\{ f = f_+ \oplus f_- \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4: \\
          & \qquad \qquad i c \alpha \cdot \nu (f_+|_\Sigma - f_-|_\Sigma) + \tfrac{1}{2}(\eta I_4 + \tau \beta) ( f_+|_\Sigma + f_-|_\Sigma) = 0 \big\}.
    \end{split}$$

In the following lemma 