On spectra of second-order differential operators

Introduction.We propose to derive some results about the differential operator (1) L = -A-Vix) on -7-dimensional Euclidean space RN from a treatment based on the theory of random processes.These methods have been developed by Kac and others; most of the results of this paper have been suggested in Reference [l].It has been shown by Rosenblatt [2] that for F(x) a non-negative function satisfying a local Lipschitz condition at almost every point of RN, the Green's function of the differential equation 1 d (2) -A*(*.t) -f (_)*(*, t) = -fix, t), t > 0, 2 at can be written in the explicit form (3) Kix, y; t) = pix -y, t)E jexp j-j F(" + *(t))_t1 | xQ) = y -x\ , where and pix, t) = Í2tI)-n'2 exp Pí.P^exp <-f Vix+ xir))dA | x{l) = y -x> denotes the average value of exp j-f Vix + xir))dr\