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.

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.

## Date:

Thu, 29/12/2016 - 13:00 to 14:00

## Location:

Ross 70