Localization for quantum graphs with a random potential

TL;DR

This paper establishes spectral localization for infinite metric graphs with a random potential by adapting multiscale analysis techniques from Euclidean spaces, under certain geometric and boundary condition assumptions.

Contribution

It extends multiscale analysis to quantum graphs with random potentials, proving localization under polynomial growth and boundary condition constraints.

Findings

Spectral localization with pure point spectrum.

Polynomial decay of eigenfunctions.

Localization occurs near the ground state energy.

Abstract

We prove spectral localization for infinite metric graphs with a self-adjoint Laplace operator and a random potential. To do so we adapt the multiscale analysis (MSA) from the R^d-case to metric graphs. In the MSA a covering of the graph is needed which is obtained from a uniform polynomial growth of the graph. The geometric restrictions of the graph include a uniform bound on the edge lengths. As boundary conditions we allow all local settings which give a lower bounded self-adjoint operator with an associated quadratic form. The result is spectral localization (i.e. pure point spectrum) with polynomially decaying eigenfunctions in a small interval at the ground state energy.