Remarks on point interactions in quantum mechanics

When dealing with singular potentials, such as point interactions, the term V(x)ψ(x) in Schrödinger's equation is, in general, mathematically ill-defined. The traditional way of dealing with this difficulty, by employing regularization techniques, has led to some ambiguous results for the δ'(x) potential in the literature. Here we propose a mathematically consistent approach to deal with the one-dimensional version of this problem by considering from the beginning the distributional nature of the whole interaction term. We show that the interaction is univocally determined from two simple mathematical requirements on the interaction distribution, together with the physical requirement of probability flux conservation and considerations of symmetry.