Mathematics

March 11-22 2024

The first week (possibly going into the second week as well) will provide an introduction to localization.

The proposed topics are:

• Frohlich-Spencer multiscale proof of large disorder almost sure localization for the random model in all dimensions. Localization is a game of resonances: with probability 1 exclude infinite chain of resonances for all energies, no tunneling to infinity.

• Spectrum of deterministic Schrodinger operators caracterized by existence of non- zero polynomially bounded solutions (Shnol’s theorem). 

• Introduction to quasi-periodic models in one dimension.

• Sketch of Frohlich-Spencer-Wittwer ’90 proof of perturbative large disorder localization for Diophantine rotations on S 1 and even, cosine-like potentials. Based on a KAM type argument and second-order perturbation theory. No analyticity, only C2.

• Analytic potentials: Bourgain-Goldstein ’00 non-perturbative localization assuming positive Lyapunov exponents. Semi-algebraic complexity bounds. Localization for systems assuming all Lyapunov exponents do not vanish (Han-S. ‘23).

• Brief excursion into Avila’s global theory: super and sub-critical cases. Almost reducibility theorem (without proof).

• Higher-dimensional quasi-periodic models. Matrix-valued Cartan theorems.

References:

• Schlag, W. An introduction to multiscale techniques in the theory of Anderson localization, Part I, Nonlinear Anal., 2022.
• Han, R. Schlag, W. Non-perturbative localization on the strip and Avila’s almost reducibility conjecture, arXiv:2306.15122
• Han, R. Schlag, W., Non-perturbative localization for quasi-periodic Jacobi block matrices, arXiv:2309.03423