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Spectral theory of deterministic and disordered systems

External Lecturer: 
W. Schlag (Yale)
Course Type: 
PhD Course
Academic Year: 
2023-2024
Period: 
March
Duration: 
20 h
Description: 

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 
Lectures schedule and location:
  •     Monday 11 March 11.00-13.00 Room 134
  •     Tuesday 12 March 11.00-13.00 Room 136
  •     Wednesday 13 March 11.00-13.00 Room 133
  •     Thursday 14 March 11.00-13.00 Room 133
  •     Friday 15 March 11.00-13.00 Room 135
  •     Monday 18 March 11.00-13.00 Room 134
  •     Tuesday 19 March 11.00-13.00 Room 136
  •     Wednesday 20 March 11.00-13.00 Room 133
  •     Thursday 21 March 11.00-13.00 Room 133
  •     Friday 22 March 11.00-13.00 Room 135


 

 

Location: 
TBC(to be checked)
Next Lectures: 
Monday, March 11, 2024 - 11:00 to 13:00
Tuesday, March 12, 2024 - 11:00 to 13:00
Wednesday, March 13, 2024 - 11:00 to 13:00
Thursday, March 14, 2024 - 11:00 to 13:00
Friday, March 15, 2024 - 11:00 to 13:00
Monday, March 18, 2024 - 11:00 to 13:00
Tuesday, March 19, 2024 - 11:00 to 13:00
Wednesday, March 20, 2024 - 11:00 to 13:00
Thursday, March 21, 2024 - 11:00 to 13:00
Friday, March 22, 2024 - 11:00 to 13:00

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