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

TL;DR

This paper investigates the spectral properties of selfadjoint Schrödinger operators on star-graphs, focusing on the multiplicity of the singular spectrum and generalizing previous results by Kac.

Contribution

It provides a new analysis of the singular spectrum's multiplicity for operators on star-graphs with standard interface conditions, extending Kac's theorem.

Findings

Derived a formula for the multiplicity of the singular spectrum in terms of Weyl functions.

Generalized Kac's theorem to a broader class of boundary relations.

Connected spectral measure properties with graph-based boundary conditions.

Abstract

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of Theorem of I.S. Kac.