We evaluate the small-amplitude excitations of a spin-polarized vapour of Fermi atoms confined inside a harmonic trap. The dispersion law ω = ω f [l + 4n(n + l + 2)/3] 1/2 is obtained for the vapour in the collisional regime inside a spherical trap of frequency ω f , with n the number of radial nodes and l the orbital angular momentum. The low-energy excitations are also treated in the case of an axially symmetric harmonic confinement. The collisionless regime is discussed with main reference to a Landau-Boltzmann equation for the Wigner distribution function: this equation is solved within a variational approach allowing an account for non-linearities. A comparative discussion of the eigenmodes of oscillation for confined Fermi and Bose vapours is presented in an Appendix.

We show that spatial Bose-Einstein condensation of non-interacting bosons occurs in dimension d < 2 over discrete structures with inhomogeneous topology and with no need of external confining potentials. Josephson junction arrays provide a physical realization of this mechanism. The topological origin of the phenomenon may open the way to the engineering of quantum devices based on Bose-Einstein condensation. The comb array, which embodies all the relevant features of this effect, is studied in detail.PACS numbers: 03.75. Fi, 85.25.Cp, The recent impressive experimental demonstration of Bose-Einstein Condensation (BEC) [1] has stimulated a new wealth of theoretical work aimed to better understanding its basic mechanisms [2] and, possibly, to exploit its consequences for the engineering of quantum devices.It is well known [3] that for an ideal gas of Bose particles BEC does not occur in dimension d ≤ 2, and an ′′ ad hoc ′′ external confining potential is needed to reach the required density of states. The same is true for free bosons living on regular periodic lattices, while the result cannot be extended to more general discrete structures lacking translational invariance.In the following we shall prove that even for d < 2 [4] non-interacting bosons may lead to Bose-Einstein condensation into a single non-degenerate state, provided one resorts to a suitable discrete non-homogeneous support structure: indeed, when the bosonic kinetic degrees of freedom do not depend on metric features only, the particles may feel a sort of effective interaction due to topology. The proposed mechanism for BEC in lower dimensional systems is then a pure effect of the structure of the ambient space and avoids as well the need of resorting to external random potentials as the ones investigated by Huang in [2]; this is a very desirable feature in view of engineering real quantum devices.In practice, the behavior of free bosons over generic discrete structures is made experimentally accessible through the realization of suitable arrays of Josephson junctions. The latter are devices that can be engineered in such a way as to realize a variety of non-homogeneous patterns. We shall show indeed that classical Josephson junction arrays arranged in a non-homogeneous geometry -not even necessarily planar -provide an example of the proposed mechanism for BEC, leading to a single state spatial condensation.Theoretical studies of Josephson junction arrays are based on the short-range Bose-Hubbard model, since the phase diagram of Josephson junction arrays may be derived [5] from an Hamiltonian describing bosons with repulsive interactions over a lattice. In d = 1 the phase diagram has been studied by analytical [6] and quantum Monte Carlo methods [7]; experimentally, Josephson junction arrays are used to study interacting bosons in one dimension. For a generic array the corresponding Hamiltonian is given bywhere A ij is the adjacency matrix: A ij = 1 if the sites i and j are nearest neighbors and A ij = 0 otherwise; a † i creates a boson at site...

In this paper we analyze the properties of electrons in noncrystalline structures, mathematically described by graphs. We consider a tight-binding model for noninteracting quantum particles and its perturbative expansion in the hopping parameter, which can be mapped into a random-walk problem on the same graph. The model is solved on a wide class of structures, called bundled graphs, which are used as models for the geometrical structure of polymers and are obtained joining to each point of a ''base'' graph a copy of a ''fiber'' graph. The analytical calculation of the Green's functions is obtained through an exact resummation of the perturbative series using graph combinatorial techniques. In particular, our result shows that when the base graph is a d-dimensional crystalline lattice, the fibers generate a self-energy of pure geometrical origin in the base Green's functions.

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