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Spectral theory of elliptic operators in exterior domains

Abstract

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m (−1)α D α a α,β (x)D β, a α,β (·) ∈ C (\( \bar \Omega \)) on smooth (bounded or unbounded) domains Ω in ℝn 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.

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Malamud, M.M. Spectral theory of elliptic operators in exterior domains. Russ. J. Math. Phys. 17, 96–125 (2010). https://doi.org/10.1134/S1061920810010085

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Keywords

  • Elliptic Operator
  • Dual Pair
  • Exterior Domain
  • Weyl Function
  • Selfadjoint Extension