Scattering theory for graphs isomorphic to a homogeneous tree at infinity

TL;DR

This paper develops a scattering theory framework for analyzing the spectral properties of graphs that resemble homogeneous trees at infinity, by reducing the problem to a finite rank perturbation and applying classical Schrödinger operator techniques.

Contribution

It introduces a novel scattering theory approach for graphs asymptotic to homogeneous trees, connecting combinatorics with spectral analysis.

Findings

Spectral theory for adjacency operators on such graphs is established.

Reduction to finite rank perturbation simplifies the analysis.

Classical Schrödinger scattering methods are adapted to graph settings.

Abstract

We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on an homogeneous tree. We developp this scattering theory using the classical recipes for Schr\"odinger operators in Euclidian spaces.