Transport and localization in random media: theory and applications
Arithmetic transitions in transport and localization for quasi-periodic operators
University of California, Irvine
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, 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.
In the singular continuous regime, the hierarchical structure of generalized eigenfunctions is responsible for unusual quantum dynamics, leading to arithmetic transitions in local dimensions and transport.
The above singular continuous results are not sharp but hold for general analytic quasiperiodic potentials. I will also describe a sharp arithmetic transition in spectral dimensions and transport holding for this general class. The talk is based on papers joint with W. Liu, S. Tcheremchantsev, and S. Zhang.