123 research outputs found
What is Localization?
We examine various issues relevant to localization in the Anderson model. We show there is more to localization than exponentially localized states by presenting an example with such states but where ⟨x(t)^2⟩/t^(2 − δ) is unbounded for any δ > 0. We show that the recently discovered instability of localization under rank one perturbations is only a weak instability
Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees
We present examples of rooted tree graphs for which the Laplacian has
singular continuous spectral measures. For some of these examples we further
establish fractional Hausdorff dimensions. The singular continuous components,
in these models, have an interesting multiplicity structure. The results are
obtained via a decomposition of the Laplacian into a direct sum of Jacobi
matrices
Absolute Continuity of the Integrated Density of States for the Almost Mathieu Operator with Non-Critical Coupling
We show that the integrated density of states of the almost Mathieu operator
is absolutely continuous if and only if the coupling is non-critical. We deduce
for subcritical coupling that the spectrum is purely absolutely continuous for
almost every phase, settling the measure-theoretical case of Problem 6 of Barry
Simon's list of Schr\"odinger operator problems for the twenty-first century.Comment: 13 pages, to appear in Inv. Mat
What is Localization?
We examine various issues relevant to localization in the Anderson model. We show there is more to localization than exponentially localized states by presenting an example with such states but where ⟨x(t)^2⟩/t^(2 − δ) is unbounded for any δ > 0. We show that the recently discovered instability of localization under rank one perturbations is only a weak instability
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
The Ten Martini Problem
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and
Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all
non-zero values of the coupling and all irrational frequencies.Comment: 31 pages, no figure
Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians
We consider the spectrum of discrete Schr\"odinger operators with Sturmian
potentials and show that for sufficiently large coupling, its Hausdorff
dimension and its upper box counting dimension are the same for Lebesgue almost
every value of the frequency.Comment: 12 pages, to appear in Commun. Math. Phy
Spectrum and diffusion for a class of tight-binding models on hypercubes
We propose a class of exactly solvable anisotropic tight-binding models on an
infinite-dimensional hypercube. The energy spectrum is analytically computed
and is shown to be fractal and/or absolutely continuous according to the value
hopping parameters. In both cases, the spectral and diffusion exponents are
derived. The main result is that, even if the spectrum is absolutely
continuous, the diffusion exponent for the wave packet may be anything between
0 and 1 depending upon the class of models.Comment: 5 pages Late
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