Subordinacy Theory on Star-Like Graphs

TL;DR

This paper extends subordinacy theory to Jacobi matrices on star-like graphs, linking solution asymptotics to spectral measure properties and analyzing singular spectrum multiplicity.

Contribution

It introduces a novel extension of subordinacy theory to star-like graphs, connecting eigenfunction behavior with spectral measure continuity.

Findings

Established a correspondence between solution asymptotics and spectral measure continuity.

Derived results on the multiplicity of the singular spectrum.

Extended subordinacy theory to a new class of graph structures.

Abstract

We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.