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Harmonic analysis on inhomogeneous amenable networks and the Bose-Einstein condensation. (English) Zbl 1362.82053
The paper is devoted to the detailed analysis of some spectral properties of the adjacency matrix (Adjacency) of a wide class of amenable inhomogeneous graphs. The obtained results are applied to the investigation of the Bose-Einstein condensation (BEC) for the Pure Hopping model (PHM). It is shown the finite volume ground state wave function converges to a weight, so the infinite volume spatial distribution of the condensate is uniquely determined. The quasi-free state considered exhibits finite local density of particles because for specific applications the unique available measurements involve finite volume regions. The paper investigates in the full generality the thermodynamics of the PHMs for considered networks relatively to the some non-trivial cases. It is proved explicitly the existence of locally normal states exhibiting BEC, provided that the Adjacency is transient, independently on the finiteness of the critical density. Such states satisfy the Kubo-Martin-Schwinger (KMS) boundary condition with respect to the natural dynamics associated with the formal Pure Hopping Hamiltonian. After recalling the standard definitions and main properties of a graph and its Adjacency, some relevant results are collected in respect to zero-density perturbation graphs. In particular, there are present the formulae for the perturbed Adjacency (the Krein Formula) and for the Laplace transform of its Integrated Density of the States. Some results involve the statistical mechanics on graphs and include some results of general nature and other concerning the PHMs and their particularization to density-zero perturbations. The facts relative to PHM include those relating to the Perron-Frobenius (PF) dimension and the Secular Equation allowing computing the norm of the perturbed Adjacency to decide whether the Hidden Spectrum appears. The PHM can exhibit BEC only at transient Adjacency. The formal Hamiltonian generates dynamics on Canonical Commutation Relations $$C^*$$-algebra containing all the Weyl unitaries and globally stable for the time evolution, so that all such states exhibiting BEC satisfy the KMS boundary condition. Then, it is discussed comb product networks for recurrent and transient situations. The PF weights and corresponding PF dimensions are computed, even for situations not exhibiting Hidden Spectrum. Then the graph and comb graphs are considered and the needed spectral properties of the corresponding Adjacency are studied in details and applied to the BEC. For sequences of finite volume chemical potentials, all the cases are covered corresponding to the condensation regime, including that corresponding to fixing amount of the condensate, and the usual one (not available for the inhomogeneous networks considered) obtained by fixing the mean density of the system.

##### MSC:
 82D55 Statistical mechanical studies of superconductors 05C10 Planar graphs; geometric and topological aspects of graph theory 42C99 Nontrigonometric harmonic analysis 46L05 General theory of $$C^*$$-algebras 46L30 States of selfadjoint operator algebras 47N50 Applications of operator theory in the physical sciences 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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