:= \int_\Sigma G_\lambda(x-y) \varphi(y) \textup{d}\sigma(y), 
  \quad \varphi \in L^2(\Sigma)^4,~x \in \mathbb{R}^3,$$ and the strongly singular boundary integral operator $\mathcal{C}_\lambda: L^2(\Sigma)^4 \rightarrow L^2(\Sigma)^4$ acting as $$\label{def_C_lambda}
  \mathcal{C}_\lambda \varphi(x) := \lim_{\varepsilon \searrow 0} \int_{\Sigma \setminus B(x, \varepsilon)} 
  G_\lambda(x-y) \varphi(y) \textup{d}\sigma(y), \quad
  \varphi \in L^2(\Sigma)^4,~x \in \Sigma,$$ where $B(x,\varepsilon)$ is the ball of radius $\varepsilon$ centered at $x$. Both operators $\Phi_\lambda$ and $\mathcal{C}_\lambda$ are well defined and bounded, see [@BEHL17 Proposition 3.4] or [@AMV15 Section 2], and $\Phi_\lambda$ is injective by [@BEHL17 Proposition 3.4 and Definition 2.3]. In particular, $\mathcal{C}_\lambda$ is uniformly bounded for $\lambda \in (-m c^2, m c^2)$, i.e. there exists a constant $K > 0$ independent of $\lambda$ such that $$\label{C_lambda_uniformly_bounded}
  \| \mathcal{C}_\lambda \| \leq K \qquad \text{ for all } \lambda \in (-m c^2,m c^2),$$ cf. [@BEHL17 Proposition 3.5] and also [@AMV15 Lemma 3.2]. Next, if $\varphi \in H^{1/2}(\Sigma)^4$, then according to [@BH17 Proposition 4.2] $$\label{equation_Phi_smooth}
  \Phi_\lambda \varphi \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4 \quad \text{and} \quad 
  \mathcal{C}_\lambda \varphi \in H^{1/2}(\Sigma)^4$$ hold. Moreover, if $\lambda \in \rho(A_0)$, then a function $f_\lambda \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4$ satisfies $$(-i c \alpha \cdot \nabla + m c^2 \beta - \lambda) f_\lambda = 0 \text{ in } \Omega_\pm,$$ if and only if there exists a 