ar X iv : 0 81 0 . 17 89 v 1 [ m at h . SP ] 1 0 O ct 2 00 8 SPECTRAL THEORY OF ELLIPTIC OPERATORS IN EXTERIOR DOMAINS

We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type A = P 06|α|,|β|6m(−1) Daα,β(x)D β , aα,β(·) ∈ C ∞(Ω), on smooth (bounded or unbounded) domains Ω in R with compact boundary ∂Ω. 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 A and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization. Our results extend classical theorems due to Vĭsik, Povzner, Birman, and Grubb.