Harmonic analysis on perturbed Cayley Trees

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.