Feynman-Kac Functional and the Schrödinger Equation

Abstract

The Feynman-Kac formula and its connections with classical analysis were inititated in [3]. Recently there has been a revival of interest in the associated probabilistic methods, particularly in applications to quantum physics as treated in [7]. Oddly enough the inherent potential theory has not been developed from this point of view. A search into the literature after this work was under way uncovered only one paper by Khas’minskii [4] which dealt with some relevant problems. But there the function q is assumed to be nonnegative and the methods used do not apparently apply to the general case; see the remarks after Corollary 2 to Theorem 2.2 below.** The case of q taking both signs is appealing as it involves oscillatory rather than absolute convergence problems. Intuitively, the Brownian motion must make intricate cancellations along its paths to yield up any determinable averages. In this respect Theorem 1.2 is a decisive result whose significance has yet to be explored. Next we solve the boundary value problem for the Schrödinger equation (Δ+2q)ϕ= 0. In fact, for a positive continuous boundary function f, a solution is obtained in the explicit formula given in (2) of §1 below, provided that this quantity is finite (at least at one point x in D). Thus the Feynman-Kac formula supplies the natural Green’s operator for the problem.