Essential self-adjointness of symmetric linear relations associated to first order systems

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. This part is based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete Riemannian manifold, $M$, and a first order differential operator, $D:\cinfz{E}\to \cinfz{F}$, acting between sections of the hermitian vector bundles $E,F$. Moreover, let $V:\cinf{E}\to L^{\infty}_{\loc}(E)$ be a self-adjoint zero order differential operator. We give a sufficient condition for the Schr\"odinger operator $H=D^tD+V$ to be essentially self-adjoint. This generalizes recent work of I. Oleinik \cite{Ole:ESA,Ole:CCQ,Ole:ESAG}, M. Shubin \cite{Shu:CQC,Shu:ESA}, and M. Braverman \cite{Bra:SAS}.