we discuss some alternative representations of the coupling condition which models the $\delta$-shell interaction:

\[lemma\_transmission\_condition\] Let $\eta, \tau \in \mathbb{R}$. Then the following hold.

-   If $\eta^2 - \tau^2 \neq -4 c^2$, then there exists an invertible matrix $R_{\eta, \tau}$ given explicitly in  such that a function $f = f_+ \oplus f_- \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4$ belongs to ${\mathrm{dom}\,}A_{\eta, \tau}$ if and only if $$f_+|_\Sigma = R_{\eta, \tau} f_-|_\Sigma.$$

-   If $\eta^2 - \tau^2 = -4 c^2$, then a function $f = f_+ \oplus f_- \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4$ belongs to ${\mathrm{dom}\,}A_{\eta, \tau}$ if and only if $$\big( 2 c I_4 - i(\alpha \cdot \nu) (\eta I_4 + \tau \beta) \big) f_+|_\Sigma = 0, \quad 
          \big( 2 c I_4 + i(\alpha \cdot \nu)  (\eta I_4 + \tau \beta) \big) f_-|_\Sigma = 0.$$

Before we prove Lemma \[lemma\_transmission\_condition\] let us discuss its meaning: if $\eta^2 - \tau^2 \neq -4 c^2$, then item (i) shows that is a coupling condition which relates the values of $f_+$ at $\Sigma$ to those of $f_-$ at $\Sigma$ via the matrix $R_{\eta, \tau}$. On the other hand, if $\eta^2 - \tau^2 = -4 c^2$, then assertion (ii) of the above lemma shows that $A_{\eta, \tau}$ is 