Spectral Theory of Elliptic Operators in Exterior Domains

2008/10/10 by Gesztesy, Fritz, Malamud, Mark M. · 1 citation
#35J40 #35P20 #47A10 #47F05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.0810.1789

Abstract

We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type \cA=∑_{0≤ |α|,|β|≤ m}(-1)^αD^αa_{α, β}(x)D^β, a_{α, β}(⋅)∈ C^∞(Ω), on smooth (bounded or unbounded) domains in \bbRⁿ 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.

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• Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

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