  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.
