Title: Sharp arithmetic spectral transitions and universal hierarchical structure of 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 signifi cant 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.