On the Continuous Spectrum of a Differential Operator COLIN CLARK Communicated by M. M. SCm~R Let G be a given region, with boundary OG, in n-dimensional Euclidean space E,. Consider the formal differential operator l[u] of Schr~Sdinger type /I-u] = -Au+q(x) u. (1) We assume that the potential function q(x) is defined and continuous on G, and bounded over any finite part of G. Let the operator L o be defined by (1) on the space ~(Lo), consisting of all infinitely differentiable complex functions on G which are restrictions to G of functions in C~ ~(E,), and which vanish on aG. In this paper we formulate a sufficient condition for the operator Lo to be semi-bounded from below, i.e. (L o u, u)>=o~ IIu II2 for some constant ~t and for all ue~(Lo). It then follows from a theorem of WEINHOLTZ-BROWDERthat Lo is essentially self-adjoint. We also obtain an estimate for the least point of the continuous spectrum of the self-adjoint extension L. The latter result contains as special cases known results of FRIEDRICHS [5] and RELLICH [9]; it also gives new information in the case G is a "quasi-finite" region and q(x) tends to -oo. Let us select a certain direction in E,; without loss of generality we may suppose it to be the direction of the positive Xt-axis. Let aG [x ~ denote the set (OG)n{x~E,: xl =x~ For x~G define p(x) = sup dist(y, 0 G [x]), (2) the supremum being taken over all ye G for which Yt = x:. Note that p (x) depends only on the first coordinate of x, and that dist(x, OG)<p(x). Our results are based on the following inequality, which partially generalizes Theorem 1 of [3]. Theorem 1. Suppose that p(x) is continuous except at most on a closed set of measure zero in G. Then for every f~ HA(G), S IVf(x)[ 2 dx> ~ [p(x)] -2 If(x)l 2 dx. (3) G G (Notation: H 1 (G) is the Sobolev space obtained by completing C~ ~(G) in the norm Ilflla={ $ (l lZf l2 + lfl2) ax}~'. G This research was supported by the United States Air Force Office of Scientific Research, under Grant number AF-AFOSR 379-65.