$$\label{equation_jump_condition_intro}
  -i c \alpha \cdot \nu (f_+|_\Sigma - f_-|_\Sigma) = \frac{1}{2} (\eta I_4 + \tau \beta) (f_+|_\Sigma + f_-|_\Sigma).$$ Therefore, the operator $A_{\eta, \tau}$ corresponding to the formal differential expression  should be defined for functions satisfying the coupling condition  and should act for $x \notin \Sigma$ as in .

The mathematical study of Dirac operators with singular potentials started in the 1980s, when Gesztesy and Šeba considered one dimensional Dirac operators with point interactions [@AGHH05; @GS87; @S89]; for more recent contributions on Dirac operators with point interactions see, e.g., [@BMP17; @CMP13; @PR14]. Based on [@GS87] and a decomposition to spherical harmonics Dittrich, Exner, and Šeba investigated the operator $A_{\eta, \tau}$ in the case that $\Sigma$ is the sphere in $\mathbb{R}^3$. In [@DES89] they showed for a wide class of parameters the self-adjointness of $A_{\eta, \tau}$ and they were able to compute its resolvent and some of its spectral properties. While some of the interesting properties of $A_{\eta, \tau}$ like the decoupling of the operator to two Dirac operators acting in $\Omega_\pm$ for interaction strengths satisfying $\eta^2 - \tau^2 = -4 c^2$ were observed in [@DES89], compare also Lemma \[lemma\_transmission\_condition\] below, others like, e.g., unexpected spectral effects for $\eta^2 - \tau^2 