14 research outputs found
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
Generic Continuous Spectrum for Ergodic Schr"odinger Operators
We consider discrete Schr"odinger operators on the line with potentials
generated by a minimal homeomorphism on a compact metric space and a continuous
sampling function. We introduce the concepts of topological and metric
repetition property. Assuming that the underlying dynamical system satisfies
one of these repetition properties, we show using Gordon's Lemma that for a
generic continuous sampling function, the associated Schr"odinger operators
have no eigenvalues in a topological or metric sense, respectively. We present
a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page
Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies
n this article we investigate numerically the spectrum of some representative
examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential
in terms of a perturbative constant b and the spectral parameter a. Our examples
include the well-known Almost Mathieu model, other trigonometric potentials with a single
quasi-periodic frequency and generalisations with two and three frequencies. We computed
numerically the rotation number and the Lyapunov exponent to detect open and collapsed
gaps, resonance tongues and the measure of the spectrum. We found that the case with one
frequency was significantly different from the case of several frequencies because the latter
has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the
spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases
with one frequency considered, gaps are always dense in the spectrum, although some gaps
may collapse either for a single value of the perturbative constant or for a range of values. In
all cases we found that there is a curve in the (a, b)-plane which separates the regions where
the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve,
which is b = 2 in the Almost Mathieu case, the measure of the spectrum is zero.Peer ReviewedPostprint (published version