On self-adjointness of a Schroedinger operator

TL;DR

This paper provides a sufficient condition for the self-adjointness of a Schrödinger operator on differential forms over a complete Riemannian manifold, extending previous scalar function results to a more general setting.

Contribution

It generalizes Oleinik's theorem by establishing self-adjointness criteria for Schrödinger operators acting on differential forms, not just scalar functions.

Findings

Established a sufficient condition for self-adjointness of the operator

Extended scalar function results to differential forms on manifolds

Generalized previous theorems to a broader geometric setting

Abstract

Let $M$ be a complete Riemannian manifold and let $\Omega^*(M)$ denote the space of differential forms on $M$. Let $d:\Omega^*(M) \to \Omega^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V(x):\Omega^*(M)\to \Omega^*(M)$ is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by Igor Oleinik about self-adjointness of a Schroedinger operator which acts on the space of scalar valued functions.