Anderson localization on quantum graphs coded by elements of a subshift of finite type

TL;DR

The paper addresses Anderson localization for Schrödinger operators on quantum graphs coded by orbits of a subshift of finite type, establishing almost-sure pure point spectrum and exponential localization. It builds a SL$(2,\mathbb{R})$ cocycle $A^E$ with Lyapunov exponent $L(E)$, proves positivity and a Uniform Large Deviations principle under a locally product structured measure, and uses holonomies and Green function estimates together with the Avalanche Principle to deduce localization. Key contributions include a finite exceptional energy set, almost-sure pure point spectrum equal to $[0,\infty)$, and exponential decay of eigenfunctions away from resonances, plus Hölder continuity of $L(E)$ on the energy interval. The results extend the ADZ2 localization framework to a quantum-graph setting, providing rigorous localization for dynamics coded by subshifts of finite type with potential implications for transport in disordered networks.

Abstract

We study Schrödinger operators on quantum graphs where the number of edges between points is determined by orbits of a "shift of finite type". We prove Anderson localization for these systems.