AUTHOR=Calçada Marcos , Lunardi José T. , Manzoni Luiz A. , Monteiro Wagner TITLE=Distributional approach to point interactions in one-dimensional quantum mechanics JOURNAL=Frontiers in Physics VOLUME=2 YEAR=2014 URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2014.00023 DOI=10.3389/fphy.2014.00023 ISSN=2296-424X ABSTRACT=

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.