13 research outputs found
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 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,) 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 -level Wegner-type estimate that measures
eigenvalue correlations through an upper bound on the probability that a local
Hamiltonian has at least 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 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.