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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, , and a
first order differential operator, D:\cinfz{E}\to \cinfz{F}, acting between
sections of the hermitian vector bundles . 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 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Braverman \cite{Bra:SAS}.Comment: Version June 2000, previous version superceded, one section adde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 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
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