Harmonic analysis on perturbed Cayley Trees

TL;DR

This paper analyzes the spectral properties of perturbed Cayley Trees, revealing insights into Bose Einstein Condensation phenomena on non-homogeneous networks with applications to Josephson junction arrays.

Contribution

It introduces a mathematical framework for studying spectral properties of adjacency operators on perturbed Cayley Trees, linking topology to Bose Einstein Condensation.

Findings

Identification of hidden spectrum below the norm

Analysis of transience character of the network

Distribution of Perron Frobenius eigenvector

Abstract

We study the mathematical aspects of the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. The resulting topological model is described by the (opposite of the) adjacency operator on the graph. In the present paper we investigate some relevant spectral properties of the adjacency of the perturbed network, such the appearance of the hidden spectrum below the norm, the transience character and the Perron Frobenius distribution. All the mentioned properties have a mathematical interest in itself, and have natural applications for the investigation of the BEC of Bardeen Cooper Bosons on the networks under consideration.