Self-adjointness of non-semibounded covariant Schr\"odinger operators on Riemannian manifolds

TL;DR

This paper investigates the self-adjointness of certain covariant Schrödinger operators on Riemannian manifolds without assuming lower semiboundedness, using elliptic and hyperbolic methods depending on a parameter.

Contribution

It establishes self-adjointness results for non-semibounded covariant Schrödinger operators on Riemannian manifolds using novel growth condition techniques.

Findings

Self-adjointness proven for operators with non-semibounded potentials.

Different methods (elliptic and hyperbolic) applied based on parameter range.

Operators satisfy self-adjointness under local Kato and Dynkin class conditions.

Abstract

In the context of a geodesically complete Riemannian manifold $M$, we study the self-adjointness of $\nabla^{\dagger}\nabla+V$ where $\nabla$ is a metric covariant derivative (with formal adjoint $\nabla^{\dagger}$) on a Hermitian vector bundle $\mathcal{V}$ over $M$, and $V$ is a locally square integrable section of $\textrm{End }\mathcal{V}$ such that the (fiberwise) norm of the "negative" part $V^{-}$ belongs to the local Kato class (or, more generally, local contractive Dynkin class). Instead of the lower semiboundedness hypothesis, we assume that there exists a number $\varepsilon \in [0,1]$ and a positive function $q$ on $M$ satisfying certain growth conditions, such that $\varepsilon \nabla^{\dagger}\nabla+V\geq -q$, the inequality being understood in the quadratic form sense over $C_{c}^{\infty}(\mathcal{V})$. In the first result, which pertains to the case $\epsilon \in [0,1)$, we use the elliptic equation method. In the second result, which pertains to the case $\varepsilon=1$, we use the hyperbolic equation method.