Coupling of symmetric operators and the third Green identity

TL;DR

This paper develops an abstract framework for the third Green identity using quasi boundary triples, and applies it to Schrödinger operators on Lipschitz domains and Riemannian manifolds.

Contribution

It introduces a novel abstract approach to the third Green identity for symmetric operators via quasi boundary triples, with applications to Schrödinger operators.

Findings

Derived an abstract form of the third Green identity for symmetric operators.

Applied the theoretical results to Schrödinger operators on Lipschitz domains.

Illustrated the approach on operators on smooth, boundaryless Riemannian manifolds.

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 $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains on smooth, boundaryless Riemannian manifolds.