114 research outputs found
Analytic quasi-perodic cocycles with singularities and the Lyapunov Exponent of Extended Harper's Model
We show how to extend (and with what limitations) Avila's global theory of
analytic SL(2,C) cocycles to families of cocycles with singularities. This
allows us to develop a strategy to determine the Lyapunov exponent for extended
Harper's model, for all values of parameters and all irrational frequencies. In
particular, this includes the self-dual regime for which even heuristic results
did not previously exist in physics literature. The extension of Avila's global
theory is also shown to imply continuous behavior of the LE on the space of
analytic -cocycles. This includes rational approximation of
the frequency, which so far has not been available
Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials
We study discrete quasiperiodic Schr\"odinger operators on \ell^2(\zee)
with potentials defined by -H\"older functions. We prove a general
statement that for and under the condition of positive Lyapunov
exponents, measure of the spectrum at irrational frequencies is the limit of
measures of spectra of periodic approximants. An important ingredient in our
analysis is a general result on uniformity of the upper Lyapunov exponent of
strictly ergodic cocycles.Comment: 15 page
Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators
We establish sharp results on the modulus of continuity of the distribution
of the spectral measure for one-frequency Schrodinger operators with
Diophantine frequencies in the region of absolutely continuous spectrum. More
precisely, we establish 1/2-Holder continuity near almost reducible energies
(an essential support of absolutely continuous spectrum). For
non-perturbatively small potentials (and for the almost Mathieu operator with
subcritical coupling), our results apply for all energies.Comment: 16 page
Delocalization in random polymer models
A random polymer model is a one-dimensional Jacobi matrix randomly composed
of two finite building blocks. If the two associated transfer matrices commute,
the corresponding energy is called critical. Such critical energies appear in
physical models, an example being the widely studied random dimer model. It is
proven that the Lyapunov exponent vanishes quadratically at a generic critical
energy and that the density of states is positive there. Large deviation
estimates around these asymptotics allow to prove optimal lower bounds on
quantum transport, showing that it is almost surely overdiffusive even though
the models are known to have pure-point spectrum with exponentially localized
eigenstates for almost every configuration of the polymers. Furthermore, the
level spacing is shown to be regular at the critical energy
Exponential dynamical localization for the almost Mathieu operator
We prove that the exponential moments of the position operator stay bounded
for the supercritical almost Mathieu operator with Diophantine frequency
Cantor spectrum of graphene in magnetic fields
We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero
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