Exactly solvable models of relativistic δ-sphere interactions in quantum mechanics

We discuss the quantum Hamiltonian HG describing a δ-sphere interaction introduced in [J. Math. Phys 30, 2275 (1989)] and formally given by HG=HD+Gδ(|x|−R), where HD is the Dirac Hamiltonian and G is a real 4×4 matrix defined by G=(0A B0). We obtain a series of new results for this model, in particular the resolvent equation, the spectral properties, the nonrelativistic limit and the various quantities related to the scattering theory. These results are generalized to the case of an asymmetric δ-sphere interaction and a δ-sphere plus Coulomb interaction, respectively.