Delta-type Dirac point interactions and their nonrelativistic limits

The problem of a relativistic free particle on a line with a hole, which is characterized in terms of boundary conditions for a one-dimensional Dirac Hamiltonian perturbed at one point, is reviewed. We show that the general four-parameter family of point interactions earlier obtained by Falkensteiner and Grosse can be written in two forms: In one of them three subfamilies of boundary conditions are obtained. In the nonrelativistic limit one of these subfamilies coincides with those given by Carreau ct al. and Carreau. Tn the other form, three subfamilies of boundary conditions are also obtained, two of which coincide with those studied by Benvegna and Dabrowski. In the nonrelativistic limit all these subfamilies coincide with those studied by Albeverio et al. The most general subfamilies for which the Dirac Hamiltonian is invariant under space inversion II as well as under time reversal T and IIT are obtained. Only these subfamilies represent delta-type Dirac point interactions. Typical relativistic and nonrelativistic boundary conditions are therein included.