%0 Journal Article
%T Distributional approach to point interactions in one-dimensional quantum mechanics
%A Marcos Cal£¿ada
%A Jos¨¦ T. Lunardi
%A Luiz A. Manzoni
%A Wagner Monteiro
%J Frontiers in Physics
%D 2014
%I Frontiers Media
%R 10.3389/fphy.2014.00023
%X 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 Schr\"odinger 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 $\delta^\prime$ interaction.
%K Quantum mechanics in one dimension
%K point interactions
%K Distribution Theory
%K $\delta$-interaction
%K $\delta '$-interaction
%U http://www.frontiersin.org/Journal/10.3389/fphy.2014.00023/abstract