Catálogo de publicaciones - libros

We present a construction of the algebra of operators and the Hilbert space for a quantum massless field in 1+1 dimensions.

We expect equilibrium states of very large systems to have a property of , called the property of return to equilibrium. This means that if the system is initially in a state that differs only a bit (say locally in space) from an equilibrium state, then it approaches that equilibrium state in the large time limit. One may view this irreversible process as a consequence of the dispersiveness of the dynamics, and the infinite size of the system: the local spatial disturbance of the equilibrium state, defining the initial state, propagates out of any bounded region if one waits long enough. Strictly speaking we expect return to equilibrium only for spatially infinitely extended systems. If one desires to observe localized events of a large, but finite systems (a laboratory) then the above description is a good approximation on an intermediate time scale (time should be large so the system can settle towards the equilibrium state, but not too large as to avoid recurrences in the finite system).

In a recent paper [17] we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain [17, 20]. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph with jump rates given by the coupling constants, i.e., the relaxation time is independent of the number of particles. Therefore, our results also provide a proof of Aldous’ Conjecture in one dimension.

We provide an introduction to the constructive results on interacting Fermions in two dimensions at thermal equilibrium above the critical temperature of pair condensation.

Let ℝ∋ξ →Σ(ξ) denote the bottom of the essential spectrum for the ber Hamiltonians of the translation invariant massive Nelson model, which describes a -dimensional electron linearly coupled to a scalar massive radiation eld. We prove that, away from a locally nite set, Σess is an analytic function of total momentum.

One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In [1] a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modeled by a periodic potential of strength ⋋. For small ⋋ and temperature Bose- Einstein condensation (BEC) is proved to occur, while at large ⋋ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect. This contribution gives a pedagogical survey of these results.

This paper is intended to constitute a review of some mathematical theories incorporating quantum corrections to the Schrödinger-Poisson (SP) system. More precisely we shall focus our attention in the electrostatic Poisson potential with corrections of power type.

We consider random Schrädinger equations on or for ≥ 3 with uncorrelated, identically distributed random potential. Denote by ⋋ the coupling constant and ψ the solution with initial data ψ.

We consider two models which exhibit equilibrium BEC superradiance. They are related to two different types of superradiant scattering observed in recent experiments. The first one corresponds to the amplification of matter-waves due to Raman superradiant scattering from a cigar-shaped BE condensate, when the recoiled and the condensed atoms are in different internal states. The main mechanism is stimulated Raman scattering in two-level atoms, which occurs in a superradiant way. Our second model is related to the superradiant Rayleigh scattering from a cigar-shaped BE condensate. This again leads to a matter-waves amplification but now with the recoiled atoms in the same state as the atoms in the condensate. Here the recoiling atoms are able to interfere with the condensate at rest to form a matter-wave grating (interference ) which has been recently observed in experiments.

We report on some recent results regarding the dynamical behavior of a trapped Bose-Einstein condensate, in the limit of a large number of particles. These results were obtained in [4], a joint work with L. Erdős and H.-T. Yau.