On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: the multi-dimensional case

Abstract

We start by pointing out an important ambiguity in the mathematical treatment of the study of bound state solutions of the Schrodinger equation for infinite well type potentials (studied for the first time in a pioneering article of 1928 by G. Gamow). An alternative to get a “localizing effect” for the wave packet solution of time dependent Schrodinger equation with potentials becoming singular on the boundary of a compact region \(\overline{\Omega }\) is here offered in terms of “ Hardy type potentials” in which the potential behaves like the distance to the boundary to the power \(\alpha =-2.\) We show that in this case the probability to find the particle outside \(\Omega \) is zero once we assume that at \(t=0\) the particle is located in \(\Omega \). The paper extends to the N-dimensional and evolution cases some previous results by the author.