via the coupling condition . Then we show for noncritical interaction strengths $\eta^2 - \tau^2 \neq 4 c^2$ the self-adjointness of $A_{\eta, \tau}$ in Theorem \[theorem\_self\_adjoint\]. In the proof of the self-adjointness we also verify a Birman-Schwinger principle, which translates the eigenvalue problem for the differential operator $A_{\eta, \tau}$ to a nonlinear eigenvalue problem for a family of integral operators acting on $\Sigma$.

In Section \[section\_spectral\_properties\] we provide the basic spectral properties of $A_{\eta, \tau}$ for noncritical interaction strengths. We compute the essential spectrum, show that at most finitely many discrete eigenvalues appear and obtain several symmetry relations for the spectrum of $A_{\eta, \tau}$. We complement the results for noncritical interactions by a theorem from [@BH17] which shows that the spectral properties of $A_{\eta, \tau}$ can be completely different in the critical case.

Finally, we compute in Section \[section\_nonrelativistic\_limit\_singular\_interaction\] for purely electrostatic and purely scalar potentials the nonrelativistic limit of $A_{\eta, \tau}$, which shows that $A_{\eta, \tau}$ is the relativistic counterpart of the Schrödinger operator $-\frac{1}{2m} \Delta + \eta \delta_\Sigma$ and which gives another justification that the jump condition  models the $\delta$-potential correctly.

Acknowledgement {#acknowledgement .unnumbered}
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The authors acknowledge financial support under the Czech-Austrian grant 7AMB17AT022 and CZ 02/2017. 