Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ_{0⩽|α|,|β|⩽m} (−1) ^α D ^α a _α,β (x)D ^β , a _α,β (·) ∈ C ^∞($ \bar \Omega $ \bar \Omega ) on smooth (bounded or unbounded) domains Ω in ℝ ⁿ with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.