Schrödinger operators with non-integer power-law potentials and Lie-Rinehart algebras

Abstract

We study Schrödinger operators \(H:= -\Delta + V\) with potentials V that have power-law growth (not necessarily polynomial) at 0 and at \(\infty \) using methods of operator algebras, microlocal analysis, and Lie theory (Lie-Rinehart algebras). More precisely, we show that H is “generated” in a certain sense by an explicit Lie algebra of vector fields (a Lie-Rinehart algebra). This allows us then to construct a suitable algebra of pseudodifferential operators that yields further properties of H by using methods of operator algebras. Classically, this method was used to study H when the power-laws describing the potential V have integer exponents. Thus, the main point of this paper is that this integrality condition on the exponents is not really necessary for the approach using pseudodifferential operators and operator algebras to work. While we consider potentials following (possibly non-integer) power-laws both at the origin and at infinity, our results extend right away to potentials having power-law singularities at several points. The extension of the classical microlocal analysis and operator algebras results to potentials with non-integer power-laws is achieved by considering the setting of Lie-Rinehart algebras and of the continuous family groupoids integrating them. (The classical case relies instead on Lie algebroids and Lie groupoids.) The algebras that we construct are useful also for the study of layer potentials.