Self-adjoint extensions of differential operators on Riemannian manifolds

TL;DR

This paper investigates conditions under which certain differential operators on Riemannian manifolds are essentially self-adjoint, extending known results to non-complete manifolds and providing criteria based on boundary behavior.

Contribution

It establishes essential self-adjointness of powers of the operator on complete manifolds and offers new sufficient conditions for non-complete cases based on boundary behavior.

Findings

Essential self-adjointness of powers of H on geodesically complete manifolds.

Sufficient boundary conditions for self-adjointness on non-complete manifolds.

Extension of self-adjointness results to broader classes of Riemannian manifolds.

Abstract

We study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.