Spectral multiplicity of selfadjoint Schroedinger operators on star-graphs with general interface conditions

TL;DR

This paper studies the spectral multiplicity of selfadjoint Schrödinger operators on star-graphs with various interface conditions, linking spectral properties to the data of decoupled operators.

Contribution

It introduces a broad class of selfadjoint interface conditions on star-graphs and analyzes their impact on spectral multiplicity, extending previous spectral theory results.

Findings

Spectral multiplicity depends on interface conditions and spectral data.

A wide class of interface conditions satisfying a generic assumption is considered.

Results connect spectral multiplicity of the coupled operator to that of decoupled operators.

Abstract

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schr\"odinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of "selfadjoint" interface conditions, subject to a assumption which is generically satisfied, is considered. We investigate properties of spectral multiplicity of singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.