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We consider the one-dimensional quantum mechanical problem of defininginteractions concentrated at a single point in the framework of the theory ofdistributions. The often ill-defined product which describes the interactionterm in the Schr\"odinger and Dirac equations is replaced by a well-defineddistribution satisfying some simple mathematical conditions and, in addition,the physical requirement of probability current conservation is imposed. Afour-parameter family of interactions thus emerges as the most general pointinteraction both in the non-relativistic and in the relativistic theories (inagreement with results obtained by self-adjoint extensions). Since theinteraction is given explicitly, the distributional method allows one to carryout symmetry investigations in a simple way, and it proves to be useful toclarify some ambiguities related to the so-called $\delta^\prime$ interaction.

Distributional approach to point interactions in one-dimensional quantum mechanics