an exterior unbounded domain is merely a mathematical object.

Another way to write the coupling condition  is $$\label{equation_transmission_condition1}
  \left( i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta) \right) f_+|_\Sigma
  + \left( -i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta) \right) f_-|_\Sigma = 0.$$ If $\eta^2 - \tau^2 \neq -4 c^2$, then the matrix $i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta)$ is invertible with $$\left(i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta)\right)^{-1} = \frac{4}{4 c^2 + \eta^2 - \tau^2} \left(- i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 - \tau \beta)\right).$$ Hence, if we set $$\label{def_R_eta}
  R_{\eta, \tau} := -\left( i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta) \right)^{-1}
      \left( - i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 + \tau \beta) \right),$$ then we deduce immediately the result of item (i). To show assertion (ii) one just has to multiply  by the matrices $\pm i c (\alpha \cdot \nu) + \frac{1}{2} (\eta I_4 - \tau \beta)$. Using  and $\eta^2 - \tau^2 = -4 c^2$ one finds that these equations simplify to the claimed boundary conditions.

Using integration 