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UID:20251227T172540EST-7439mhZxNV@132.216.98.100
DTSTAMP:20251227T222540Z
DESCRIPTION:Sharp arithmetic transitions for 1D quasiperiodic operators\n\n
 A very captivating question in solid state physics is to determine/underst
 and the hierarchical structure of spectral features of operators describin
 g 2D Bloch electrons in perpendicular magnetic fields\, as related to the 
 continued fraction expansion of the magnetic flux. In particular\, the hie
 rarchical behavior of the eigenfunctions of the almost Mathieu operators\,
  despite significant numerical studies and even a discovery of Bethe Ansat
 z solutions has remained an important open challenge even at the physics l
 evel.\n	\n	I will present a complete solution of this problem in the exponen
 tial sense throughout the entire localization regime. Namely\, I will desc
 ribe the continued fraction driven hierarchy of local maxima\, and a unive
 rsal (also continued fraction expansion dependent) function that determine
 s local behavior of all eigenfunctions around each maximum\, thus giving a
  complete and precise description of the hierarchical structure. In the re
 gime of Diophantine frequencies and phase resonances there is another univ
 ersal function that governs the behavior around the local maxima\, and a r
 eflective-hierarchical structure of those\, phenomena not even described i
 n the physics literature. These results lead also to the proof of sharp ar
 ithmetic transitions between pure point and singular continuous spectrum\,
  in both frequency and phase\, as conjectured since 1994. This part of the
  talk is based on the papers joint with W. Liu.Within the singular continu
 ous regime\, it is natural to look for further\, dimensional transitions. 
 I will present a sharp arithmetic transition result in this regard that ho
 lds for the entire class of analytic quasiperiodic potentials\, based on t
 he joint work with S. Zhang.\n
DTSTART:20181116T210000Z
DTEND:20181116T220000Z
LOCATION:Room 1140\, CA\, Pav. André-Aisenstadt
SUMMARY:Svetlana Jitomirskaya\, UC Irvine / Chaire Aisenstadt 2018
URL:/mathstat/channels/event/svetlana-jitomirskaya-uc-
 irvine-chaire-aisenstadt-2018-291500
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