Analytical approach to the wave function of a decaying quantum system

Recently we outlined an analytical method of Solving the time-dependent Schrödinger equation for a model which simulates a decaying quantum system such as an α-decaying nucleus. A particle in the model is initially confined around the origin and leaks out, tunneling through a potential barrier. The solution can be expressed as a linear combination of the Moshinsky functions, each of which is associated with a pole of the Scattering S matrix of the model. In this paper we give a full account of the method with a few explicit examples. We examine deviations from the exponential decay law at very large times. We comment on a recent controversy regarding the t dependence of the survival and nonescape probabilities when t is very large.