Semiclassical Approximation Using Wiener Measure

Abstract

To represent the wave function for the Schrödinger equation by a path integral raises the question of knowing if the path integral considered is well-defined. That is, given any potential V, is the associated oscillatory function path-integrable? This question is proved to be irrelevant regarding the semiclassical approximation problem. Indeed a probabilistic ansatz based on Wiener measure is exhibited, which under mild conditions on the potential, leads to the semiclassical approximation before the caustics. Such an anstaz is available after the caustics, but the link between this ansatz and the wave function is an open problem.