A probabilistic approach to semi-classical approximations (Q1911524)

The paper deals with semi-classical approximation (Planck constant going to zero) of the Schrödinger equation. A new probabilistic approach based on the Wiener measure is proposed. The paper is in the line of the method of analytic extension of the Feynman-Kac formula, as opposite to the approach based on a Feynman integration. Under strong analyticity assumptions the analytic extension is known to be successfull. The main point of the paper is the development of an almost-analytic extension, based on recent results on related techniques, in order to avoid analyticity assumptions on the potentials. The almost-analytic extension does not give an exact probabilistic representation of the wave function; however, a probabilistic ansatz can be given, which approximates very well the wave function in the semi-classical regime. The semi-classical expansion is obtained under mild hypotheses on the potential at infinity, without analyticity conditions. The paper is limited to short times before caustics.