Distributional approach to point interactions in one-dimensional quantum mechanics

*Correspondence: José T. Lunardi, Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Avenida General Carlos Cavalcanti, 4748, 84030-900 Ponta Grossa, PR, Brazil e-mail: [email protected] We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called δ′ interaction.