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M. Shubin \cite{Shu:CQC,Shu:ESA}, and M. Braverman \cite{Bra:SAS}.Comment: Version June 2000, previous version superceded, one section adde

On the Determinant of One-Dimensional Elliptic Boundary Value Problems

We discuss the $\zeta-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.Comment: LaTeX, 18 page

Regularizing infinite sums of zeta-determinants

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm-Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the zeta-determinant of an infinite direct sum of Sturm-Liouville operators. The Laplace-Beltrami operator on a surface of revolution decomposes into an infinite direct sum of Sturm-Louville operators, parametrized by the eigenvalues of the Laplacian on the cross-section. We apply the polyhomogeneous expansion to equate the zeta-determinant of the Laplace-Beltrami operator as a regularized sum of zeta-determinants of the Sturm-Liouville operators plus a locally computable term from the polyhomogeneous resolvent trace asymptotics. This approach provides a completely new method for summing up zeta-functions of operators and computing the meromorphic extension of that infinite sum to s=0. We expect out method to extend to a much larger class of operators