Smoothness of Correlations in the Anderson Model at Strong Disorder

We study the higher-order correlation functions of covariant families of observables associated with random Schr\"odinger operators on the lattice in the strong disorder regime. We prove that if the distribution of the random variables has a density analytic in a strip about the real axis, then these correlation functions are analytic functions of the energy outside of the planes corresponding to coincident energies. In particular, this implies the analyticity of the density of states, and of the current-current correlation function outside of the diagonal. Consequently, this proves that the current-current correlation function has an analytic density outside of the diagonal at strong disorder

Dynamical Systems on Spectral Metric Spaces

Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak*-topology on the state space of A. A *-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra AxZ, characterizing the metric properties of the dynamical system ? If $\alpha$ is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,$\alpha$) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this new system are discussed. Applications, in number theory, in coding theory are given at the end

Correlations Estimates in the Lattice Anderson Model

We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schr\"odinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new $n$-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least $n$ eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.Comment: 16 page

Two interacting Hofstadter butterflies

The problem of two interacting particles in a quasiperiodic potential is addressed. Using analytical and numerical methods, we explore the spectral properties and eigenstates structure from the weak to the strong interaction case. More precisely, a semiclassical approach based on non commutative geometry techniques permits to understand the intricate structure of such a spectrum. An interaction induced localization effect is furthermore emphasized. We discuss the application of our results on a two-dimensional model of two particles in a uniform magnetic field with on-site interaction.Comment: revtex, 12 pages, 11 figure

Random Fibonacci Sequences

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent lambda(B) in powers of B and B^{-1}, respectively. For the classical case of $\beta=1$ we obtain exact non-perturbative results. In particular, an invariant measure associated with Ricatti variable r_n=x_{n+1}/x_{n} is shown to exhibit plateaux around all rational.Comment: 11 Pages (Multi-Column); 3 EPS Figures ; Submitted to J. Phys.

Theory of aperiodic solids

Issued as final reportNational Science Foundatio