Coupling of symmetric operators and the third Green identity

Abstract

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension T of a symmetric operator S in a Hilbert space H, employing the technique of quasi boundary triples for T . The general

Boundary Value Problems, Weyl Functions, and Differential Operators

“…The Kreȋn-von Neumann extension which is of special interest in this context was investigated in, e.g., [14,68,69,70,71,93,181,356,362,578,579]. For coupling methods for elliptic differential operators based on boundary triplet techniques in the spirit of Section 8.6 we refer to the recent paper [88], where also an abstract version of the third Green identity was proved. Finally, we mention that various classes of λ-dependent boundary value problems for elliptic operators can be treated in such a context, see, e.g., [87,103,137,285,286].…”

On Compressed Resolvents of Schrödinger Operators with Complex Potentials

“…In the self-adjoint context it also follows from abstract operator theory principles that the compressed resolvent (1.2) can be described via the Kreȋn-Naȋmark formula or can be seen as aŠtraus family of extensions of a symmetric operator in L 2 (Ω), see, e.g., [3,Chapter 2.7], the contributions [4,11,12,13,14,20,21] by Henk de Snoo and his coauthors, and also the classical works [22,23,24,25,31]. However, it is of particular interest to determine the various operators and mappings that appear in the classical abstract Kreȋn-Naȋmark formula for the present case of a Schrödinger operator; for real potentials V an explicit expression for the compressed resolvent (1.2) was given in [3,Theorem 8.6.3] and for Lipschitz subdomains of Riemann manifolds in [1,Corollary 5.5].…”