Radial bounds for Schrödinger operators in euclidean domains

Abstract

We extend study of regularity properties of elliptic operators (c.f. [2]) to second order operators on domains bounded by finite numbers of hyperplanes. Previous results for Euclidean space and the symmetry of the domains are exploited to obtain resolvent bounds. Corollaries include semigroup generation, essential self-adjointness, and regularity of eigenfunction expansions for such operators. The present work provides basic results aimed at extending regularity information for partial differential operators (especially with singular coefficients) to a general class of operators in domains with boundary. In one dimension these results encompass a body of work in Sturm-Liouville theory on the half-line.