Spectral theory of elliptic operators in exterior domains
By Fritz Gesztesy, M. Malamud. 2008
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AbstractDiverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m(−1)αDαaα,β(x)Dβ, aα,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.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f1e9d692289b20858672aba666482df3f1bc5c59
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Citations in the corpus
- Square‐integrable solutions and Weyl functions for singular canonical systems (2011, background): “Furthermore, the reader is referred to [5]–[7], [9], [10], [12], [20], [21], [30], [53], [55] for some recent extensions and applications of the concept of boundary triplets and their Weyl functions.”
- Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, background): “…and Trunk, 2007; Brasche, Malamud and Neidhardt, 2002; Brüning, Geyler and Pankrashkin, 2008; Derkach, Hassi and de Snoo, 2003; Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Karabash, Kostenko and Malamud, 2009; Kostenko and Malamud, 2010; Posilicano, 2008] and the references therein.”
- Boundary pairs associated with quadratic forms (2012, background): “perator (Theorems 4.9 and 4.18), namely λ∈ σ(H) iff 0 ∈ σ(Λ(λ)), (1.2) provided λ/∈ σ(HD). This spectral relation extends known results for (ordinary) boundary triples to our general context (see e.g. [M10], [BGW09, Cor. 2.3, Sec. 2.2], [BGP08, BMN02], going back to ideas already contained in [V52] a”