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Sharp arithmetic spectral transitions and universal hierarchical structure of quasiperiodic eigenfunctions

Connections for Women: Harmonic Analysis January 19, 2017 - January 20, 2017

January 20, 2017 (11:00 AM PST - 12:00 PM PST)
Speaker(s): Svetlana Jitomirskaya (University of California, Irvine)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • quasiperiodic

  • eigenfunctions

  • spectral transitions

  • universal hierarchical structure

  • harmonic analysis

  • PDE

  • mathematical physics

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Quasiperiodic Eigenfunctions

Abstract

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level.

I will present a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, I will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature.

These results lead also to the proof of sharp arithmetic transitions between pure point and singular continuous spectrum, in both frequency and phase, as conjectured since 1994. The talk is based on papers joint with W. Liu

Supplements
27801?type=thumb Jitomirskaya Notes 3.42 MB application/pdf Download
Video/Audio Files

Quasiperiodic Eigenfunctions

H.264 Video 07-Jitomirskaya.mp4 2.25 GB video/mp4 rtsp://videos.msri.org/data/000/027/647/original/07-Jitomirskaya.mp4 Download
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