Spectral Theory of Elliptic Operators in Exterior Domains

Details Spectral Theory of Elliptic Operators in Exterior Domains Spectral Theory of Elliptic Operators in Exterior Domains

2008-10-10

arxiv.org/abs/0810.1789v1

Mathematics

Spectral Theory

6 pages

Scientific paper

We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type $\cA=\sum_{0\le |\alpha|,|\beta|\le m}(-1)^\alpha D^\alpha a_{\alpha, \beta}(x)D^\beta$, $a_{\alpha, \beta}(\cdot)\in C^{\infty}({\overline\Omega})$, on smooth (bounded or unbounded) domains in $\bbR^n$ with compact boundary. Using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of these closed realizations of $\cA$ and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization. Our results extend classical theorems due to Visik, Povzner, Birman, and Grubb.

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