Schr\"odinger operators with non-integer power-law potentials and Lie-Rinehart algebras

Abstract

We study Schr\"odinger operators $H:= -\Delta + V$ with potentials $V$ that have power-law growth (not necessarily polynomial) at 0 and at $\infty$ using methods of Lie theory (Lie-Rinehart algebras) and microlocal analysis. More precisely, we show that $H$ is ''generated'' in a certain sense by an explicit Lie-Rinehart algebra. This allows then to construct a suitable (microlocal) calculus of pseudodifferential operators that provides further properties of $H$. Classically, this microlocal analysis 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 microlocal analysis method 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 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.)