Cantor spectrum for one-frequency quasi-periodic
Schrödinger operators
October 20, 2014
1:00 pm PHG 646
Abstract
The spectrum of one-frequency quasi-periodic Schrödinger operators are
widely expected to be Cantor. However, it turns out that `Cantor or
non-Cantor', and the mechanism leads to Cantor depend strongly on the
regularity of the potentials. After reviewing some recent works, I will
first present a joint work with Artur Avila and David Damanik where we
constructed a class of \(C^0\) potentials with non-Cantor spectrum, which
are the first examples of this kind. Then I will show a recent joint work
with Yiqian Wang where we obtained for a class of \(C^2\) potentials and
for any fixed Diophantine frequency, the spectrum is Cantor. This
is the first rigorous result of this type for quasi-periodic potentials
beyond the \(C^0\) and the real analytic categories.
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Last modified: April 08 2016 - 20:30:35