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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:One-dimensional quasi-periodic Schrdinger operator
s I - Marx\, C (Oberlin College)
DTSTART;TZID=Europe/London:20150113T113000
DTEND;TZID=Europe/London:20150113T123000
UID:TALK56973AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/56973
DESCRIPTION:Quasi-periodic Schrdinger operators arise in solid
state physics\, describing the influence of an ex
ternal magnetic field on the electrons of a crysta
l. In the late 1970s\, numerical studies for the m
ost prominent model\, the almost Mathieu operator
(AMO)\, produced the first example of a fractal in
physics known as "Hofstadter's butterfly\," marki
ng the starting point for the on-going strong inte
rest in such operators in both mathematics and phy
sics. \nWhereas research in the first three decade
s was focussed mainly on unraveling the unusual pr
operties of the AMO\, in recent years a combinatio
n of techniques from dynamical systems with those
from spectral theory has allowed for a more "globa
l\," model-independent point of view. Intriguing p
henomena first encountered for the AMO\, notably t
he appearance of a critical phase corresponding to
purely singular continuous spectrum\, could be te
sted for their prevalence in general models. \nThi
s workshop will introduce the participants to some
of the techniques necessary to understand the spe
ctral properties of quasi-periodic Schrdinger oper
ators. The presentation is of expository nature an
d will particularly emphasize the close ties to dy
namical systems (``matrix cocycles'')\, which was
successfully used to address several open problem
s (e.g. the ``Ten Martini problem'') and enabled a
bove-mentioned global perspective.\n\nTopics inclu
ded are: \nBasics: collectivity and regularity of
spectral properties\, matrix cocycles\, arithmetic
conditions \nLyapunov exponent: positivity and co
ntinuity \nSupercritical behavior - Anderson loca
lization \nSubcritical behavior - Duality\, reduci
bility\, and absolutely continuous spectrum \nAvil
a's global theory and critical behavior \n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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