The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

Abstract

The Krein–von Neumann extension is studied for Schrödinger 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.

Spectral Monotonicity for Schrödinger Operators on Metric Graphs

“…Many properties of the non-trivial behavior of Schrödinger operators on metric graphs can be investigated in terms of spectral theory, which has been studied in recent years in various perspectives such as spectral estimates, properties of eigenfunctions, inverse problems or questions of isospectrality, see [1,2,9,12,14,18,22,23,24,34,35,39,40,52,57,58,60,61] for a few of the most recent developments.…”

“…In recent years, the effect of geometric manipulations of the metric graph onto the eigenvalues of Schrödinger operators have received much attention. This is mainly due to the fact that such so-called surgery principles have turned out to be powerful tools for obtaining eigenvalue bounds and prove isoperimetric inequalities [3,8,9,14,15,33,49,51,53,56,58,63,64].…”