The Spectrum of a Magnetic Schr\"odinger Operator with Randomly Located Delta Impurities

We consider a single band approximation to the random Schroedinger operator in an external magnetic field. The spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice. In this paper we generalize these results by letting the delta impurites have random positions as well as strengths; they are located in squares of a lattice with a general bounded distribution. We characterize the entire spectrum of this operator when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level is infinitely degenerate and that the eigenfunctions corresponding to other Landau band energies are exponentially localized.Comment: 38 pages, LaTeX2e, macros included (to appear in J. Math. Phys.

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schro\"dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization.Comment: To appear in Commun. Math. Phys. (1999

The statistics of the grand canonical number density for interacting bosons

It is shown that the weak law of large numbers holds for the grand canonical number density in a system of bosons interacting through a pair potential which is superstable. An estimate of the probability of large deviations is obtained in terms of the canonical free energy density

A Model of Continuous Polymers with Random Charges

We study a model of polymers with random charges; the possible shapes of the polymer are represented by the sample paths of a Brownian motion, and the cumulative charge distribution along a polymer is modelled by a realisation of a Brownian bridge. Charges interact through a general positive-definite two-body potential. We study the infinite volume free energy density for a fixed realisation of the Brownian motion; this is not self-averaging, but shows on the contrary a sample dependence through the local time of the Brownian motion. We obtain an explicit series representation for the free energy density; this has a finite radius of convergence, but the free energy is nevertheless analytic in the inverse temperature in the physical domain

Calculation of the Invariant Measures at Weak Disorder for the Two-Line Anderson Model

We compute the invariant measures in the weak disorder limit, for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterise the measures. It turns out that at zero energy, there is a similar anomaly as first found by Kappus and Wegner for the single chain, but that, in addition, the measures take a different form on different regions of the spectrum

A Pair Hamiltonian Model of a Non-ideal Boson Gas

The pressure in the thermodynamic limit of a non-ideal Boson gas whose Hamiltonian includes only diagonal and pairing terms can be expressed as the infimum of a functional depending on two measures on momentum space: a positive measure describing the particle density and a complex measure describing the pair density. In this paper we examine this variational problem with the object of determining when the model exhibits Bose-Einstein condensation. In addition we show that if the pairing term in the Hamiltonian is positive then it has no effect

Invariant Measures for One-Dimensional Anderson Localisation

We compute the invariant measures for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterise the measures. It turns out that there is a similar anomaly as first found by Kappus and Wegner for the single chain, but that, in addition, the measures take a different form on different regions of the spectrum

The Large Deviation Principle for the Kac Distribution

We prove that the Large Deviation Principle holds for the distribution of the particle number density (the Kac distribution) whenever the free energy density exists in the thermodynamic limit. We use this result to give a new proof of the Large Deviation Principle for the Kac distribution of the free Boson gas. In the case of mean-field models, non-convex rate functions can arise; this is illustrated in a model previously studied by E.B. Davies

A Dicke Type Model for Equilibrium BEC Superradiance

We study the effect of electromagnetic radiation on the condensate of a Bose gas. In an earlier paper we considered the problem for two simple models showing the cooperative effect between Bose-Einstein condensation and superradiance. In this paper we formalise the model suggested by Ketterle et al in which the Bose condensate particles have a two level structure. We present a soluble microscopic Dicke type model describing a thermodynamically stable system. We find the equilibrium states of the system and compute the thermodynamic functions giving explicit formulae expressing the cooperative effect between Bose-Einstein condensation and superradiance