The Krein-von Neumann extension for Schr\"odinger operators on metric graphs

TL;DR

This paper investigates the Krein-von Neumann extension for Schrödinger operators on metric graphs, providing explicit vertex conditions, spectral characterizations, and analyzing how eigenvalues change under graph perturbations.

Contribution

It explicitly characterizes the Krein-von Neumann extension's vertex conditions and explores its spectral properties and perturbation behavior on metric graphs.

Findings

Explicit vertex conditions for the Krein-von Neumann extension.

Variational characterization of positive eigenvalues.

Eigenvalue inequalities and perturbation analysis.

Abstract

The Krein-von Neumann extension is studied for Schr\"odinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.