-δ plus a bad potential

The purpose of this paper is to associate with -Δ+V(x) a self-adjoint operator in case V is nonnegative. This has been done in many ways by previous authors under a variety of conditions. The point of the present work is to do this under conditions applying to, e.g., any many-body potential, the only requirements being that V:R ³ⁿ→[0,∞] be continuous and that V^-1(∞) be not too large in the sense of volume and/or capacity; especially, no conditions of growth are imposed upon V. The method of proof is probabilistic, being based upon familiar properties of the Brownian motion.