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SUMMARY:Diophantine properties and the spectral theory of explicit quasipe
 riodic models - Jitomirskaya\, S (University of California\, Irvine)
DTSTART:20150324T100000Z
DTEND:20150324T110000Z
UID:TALK58546@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:The development of the spectral theory of quasiperiodic operat
 ors has been largely centered around and driven by several explicit models
 \, all coming from physics. In this talk we will review the highlights of 
 the current state-of-the-art of the spectral theory for the following thre
 e models: almost Mathieu operator\, extended Harper's model and Maryland m
 odel\, focusing on arithmetically driven spectral transitions\, measure of
  the spectrum\, and the Cantor nature of the spectrum.\n Those models all 
 demonstrate interesting dependence on the arithmetics of parameters (even 
 in some cases when the final conclusion does not have such dependence) and
  have traditionally been approached through KAM-type schemes. Even when th
 e KAM arguments have been replaced by the non-perturbative ones allowing t
 o treat more couplings\, frequencies that are neither far from nor close e
 nough to rationals presented a challenge as for them there was nothing lef
 t to perturb about. A remarkable relatively recent development concerning 
 the explicit models is that very precise results have become possible: not
  only many facts have been established for a.e. frequencies and phases\, b
 ut in many cases it has become possible to go deeper in the arithmetics an
 d either establish precise arithmetic transitions or even obtain results f
 or all values of parameters. More detailed talks on some of the covered to
 pics will be given in the April workshop by J. You\, Q. Zhou\, R. Han\, an
 d W. Liu.
LOCATION:Seminar Room 1\, Newton Institute
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