Spectrum of the Static Potential Schrodinger Equation over E n

Abstract

over x e En, Xeuclidean n space, for u E L2(E,), where V2 = 82/8x1 + * * + a2/8x2, i3 is the spectral parameter, and V(x) is a prescribed real measurable function. The basic spectral theory of (1.1) for the half line, u e L2(O, o ) with a boundary condition at x = 0, has been developed by Weyl [1] and amplified by Titchmarsh [2] and the recent work of Wintner and others, [3]-[6], in the American Journal. The results here extend easily to the whole line, u e L2(E1), but for L2(En) with n > 1 the theory, [8]-[15], is much less complete, particularly results on the analysis of the spectrum. In all these papers fairly strong local regularity conditions are imposed on V(x) in order that (1.1) may have meaning. However, by using a construction suggested by p. 429 of Carleman's paper [8], we can get a Green's function for (1.1) under only a very mild global integrability condition on V(x), specifically I V 1a, < 1 for some large a, where as in (3.2)