HARMONIC ANALYSIS ON CAYLEY TREES II: THE BOSE–EINSTEIN CONDENSATION

Abstract

We investigate the Bose–Einstein Condensation on non-homogeneous non-amenable networks for the model describing arrays of Josephson junctions. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, has also a mathematical interest in itself. The present paper is then the application to the Bose–Einstein Condensation phenomena, of the harmonic analysis aspects, previously investigated in a separate work, for such non-amenable graphs. Concerning the appearance of the Bose–Einstein Condensation, the results are surprisingly in accordance with the previous ones, despite the lack of amenability. The appearance of the hidden spectrum for low energies always implies that the critical density is finite for all the models under consideration. We also show that, even when the critical density is finite, if the adjacency operator of the graph is recurrent, it is impossible to exhibit temperature states which are locally normal (i.e. states for which the local particle density is finite) describing the condensation at all. A similar situation seems to occur in the transient cases for which it is impossible to exhibit locally normal states ω describing the Bose–Einstein Condensation with mean particle density ρ(ω) strictly greater than the critical density ρc. Indeed, it is shown that the transient cases admit locally normal states exhibiting Bose–Einstein Condensation phenomena. In order to construct such locally normal temperature states by infinite volume limits of finite volume Gibbs states, a careful choice of the sequence of the chemical potentials should be done. For all such states, the condensate is essentially allocated on the base point supporting the perturbation. This leads to ρ(ω) = ρc as the perturbation is negligible with respect to the whole network. We prove that all such temperature states are Kubo–Martin–Schwinger states for the natural dynamics associated to the (formal) pure hopping Hamiltonian. The construction of such a dynamics, which is a delicate issue, is also provided in detail.