Abstract
The problem of a relativistic free particle on a line with a hole, which ischaracterized in terms of boundary conditions for a one-dimensional DiracHamiltonian perturbed at one point, is reviewed. We show that the generalfour-parameter family of point interactions earlier obtained by Falkensteiner andGrosse can be written in two forms: In one of them three subfamilies of boundaryconditions are obtained. In the nonrelativistic limit one of these subfamiliescoincides with those given by Carreau et al. and Carreau. In the other form, threesubfamilies of boundary conditions are also obtained, two of which coincide withthose studied by Benvegnu;ag and Dabrowski. In the nonrelativistic limit all thesesubfamilies coincide with those studied by Albeverio et al. The most generalsubfamilies for which the Dirac Hamiltonian is invariant under space inversionΦP as well as under time reversal T and PΦT are obtained. Only these subfamiliesrepresent delta-type Dirac point interactions. Typical relativistic andnonIrelativistic boundary conditions are therein included.
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