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In this article, Dirac operators \(A_{\eta , \tau }\) coupled with combinations of electrostatic and Lorentz scalar \(\delta \)-shell interactions of constant strength \(\eta \) and \(\tau \), respectively, supported on compact surfaces \(\Sigma \subset \mathbb {R}^3\) are studied. In the rigorous definition of these operators, the \(\delta \)-potentials are modeled by coupling conditions at \(\Sigma \). In the proof of the self-adjointness of \(A_{\eta , \tau }\), a Krein-type resolvent formula and a Birman–Schwinger principle are obtained. With their help, a detailed study of the qualitative spectral properties of \(A_{\eta , \tau }\) is possible. In particular, the essential spectrum of \(A_{\eta , \tau }\) is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of \(A_{\eta , \tau }\) is computed and it is discussed that for some special interaction strengths, \(A_{\eta , \tau }\) is decoupled to two operators acting in the domains with the common boundary \(\Sigma \).

On Dirac operators in $$\mathbb {R}^3$$ R 3 with electrostatic and Lorentz scalar $$\delta $$ δ -shell interactions