Workshop on Dynamical Systems and Aperiodic Order
Bielefeld University, Germany
14th - 17th March 2011
All talks take place in Room V3-201 ("Common Room") in the Bielefeld
University main building.
Click here for
a description of the main building.
Dynamical bounds for Sturmian operators
Laurent Marin
Abstract:
We study the dynamics of discrete, one-dimensional, sturmian Schrödinger
operators.
We consider
H = Δ +
V, where Δ is the discrete Laplacian and
V is a discrete quasiperiodical function associated to a rotation
irrationnal number.
The main result is a dynamical bound from above for transport exponents
that valuate speed of the wavepacket spreading under time evolution. This
bound is true for almost every sturmian potential and is sub-ballistic for a
coupling constant big enough. This bound is valid with respect to a full
Lebesgue mesure diophantine condition on the irrational number associated to
the potential. This condition is true for almost every irrational numbers.
We show an example of irrational number with ballistic motion at any
coupling constant. We study the fractal dimension of the spectrum of these
operators which can bound from below, under more restrictive assumptions,
transport exponents. We get a new bound from below for the box dimension of
the spectrum.
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