Self-adjoint extensions of differential operators on Riemannian manifolds
Résumé
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 $M$.
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