Path integrals and the essential self-adjointness of differential operators on noncompact manifolds

TL;DR

This paper proves that Schrödinger operators on complete noncompact Riemannian manifolds are essentially self-adjoint using path integral methods, with applications to quantum operators like the Pauli-Dirac operator on 3-manifolds.

Contribution

It introduces a path integral approach to establish essential self-adjointness of differential operators on noncompact manifolds, extending previous results to more general settings.

Findings

Operators are essentially self-adjoint under geodesic completeness.

Operator closures are semibounded from below.

Results apply to quantum operators on Riemannian 3-manifolds.

Abstract

We consider Schr\"odinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on the space of smooth functions with compact support, and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli-Dirac operators that describe the energy of Hydrogen type atoms on Riemannian 3-manifolds.