Localization on quantum graphs with random vertex couplings

TL;DR

This paper investigates localization phenomena in quantum graphs with random vertex couplings, establishing conditions under which localization occurs, especially in strong disorder and at spectral edges, using self-adjoint extension techniques.

Contribution

It introduces finite volume criteria for localization on quantum graphs with random couplings, expanding understanding of disorder effects in quantum graph models.

Findings

Localization conditions are derived for strong disorder.

Localization occurs at spectral edges under certain conditions.

Method applies to a class of periodic quantum graphs.

Abstract

We consider Schr\"odinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.