A geometric approach to absolutely continuous spectrum for discrete Schr\"odinger operators

TL;DR

This paper reviews a geometric method for establishing the absolutely continuous spectrum in Schrödinger operators, covering decaying potentials, Anderson models on trees, and introducing loop tree models with open questions.

Contribution

It presents a simplified proof for ac spectrum in the Anderson model on trees and introduces new loop tree models, expanding the understanding of spectral properties in these systems.

Findings

Proved ac spectrum for Anderson model on trees

Extended ac spectrum results to percolation models on trees

Introduced novel loop tree models with open research problems

Abstract

We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schr\"odinger operators developed in \cite{FHS1,FHS2,FHS3,FHS4}. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems.