Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

TL;DR

This paper proves that the absolutely continuous spectrum of the Laplacian on a rooted metric tree graph remains stable under weak random perturbations of edge lengths, extending spectral analysis techniques to tree structures.

Contribution

It introduces a generalized Weyl-Titchmarsh function for trees and demonstrates spectral stability under weak disorder, a novel approach in quantum graph analysis.

Findings

Absolutely continuous spectrum is stable under weak disorder.

Introduces a directional transmission amplitude function for trees.

Provides bounds on fluctuations of the transmission function.

Abstract

We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 2 $ and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.