Spectral Theory of Elliptic Operators in Exterior Domains
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We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type $\cA=\sum_{0\le |α|,|β|\le m}(-1)^αD^αa_{α, β}(x)D^β$, $a_{α, β}(\cdot)\in C^{\infty}({\overlineΩ})$, 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.
6 pages
6 pages