The University of Arizona

Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators

Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators

Series: Special Mathematics Colloquium
Location: Math 501
Presenter: Wencai Liu, Irvine

The Harper's model is a tight-binding description of Bloch electrons on 
$\mathbb{Z}^2$ under a constant transverse magnetic field.
In 1964, Mark Azbel predicted that both spectra and eigenfunctions of 
this model
  have self-similar hierarchical structure driven by the continued 
fraction expansion of the irrational magnetic flux.
In 1976, the hierarchical structure of spectra was discovered 
numerically by Douglas Hofstadter, and was later observed in various 
experiments. The mathematical study of Harper's model led to the 
development of spectral theory of the almost Mathieu operator, with the 
solution of the Ten Martini Problem partially confirming the fractal 
structure of the spectrum.

In this talk we will present necessary background and discuss the main 
ideas behind our confirmation (joint with S. Jitomirskaya) of  Azbel's 
second prediction of the structure of the eigenfunctions. More 
precisely, we show that the eigenfunctions of the almost Mathieu 
operators in the localization regime, feature self-similarity governed 
by the continued fraction expansion of the frequency. These results also 
lead to the proof of sharp arithmetic transitions between pure point and 
singular continuous spectra, both in the frequency and the phase, as 
conjectured in 1994.

The talk will be intended for a general audience.  Basic knowledge of 
continued fraction expansions and the spectral theory of self-adjoint 
operators will be helpful, but not necessary.

Refreshments in Math Commons Room at 3:30pm

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