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.
Constructive gap-labeling and scaling properties of Schrödinger operator with Sturmian potential
Laurent Raymond
Abstract:
We consider the class of quasiperiodic Schrödinger operator with
so-called step-potential. By means of the trace map, we describe the
spectrum by putting it in one-to-one correspondence with a Cantor set
of sequences of symbols.
This symbolic representation allows a computation of the integrated
density of states for all energies of the spectrum. The identification
of the gap-edges leads to the gap labelling of the operator. Namely,
the set of gaps of the spectrum is in one-to-one correspondence with
the set of relative integers. We also get some insight on the scaling
properties of the bands and gaps under the action of the trace-map.
To investigate the dynamical properties of these operators, such as
the time-evolution of a wave-packet, a better description of the
spectral measure is needed. As a starting point, the golden-mean case,
is a good candidate.
In this so-called Fibonacci model, we obtain an upper bound for the
Haussdorff dimension of the spectrum. To go one step further, a
numerical inversion of the coding function is performed and some
scaling properties are shown.
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