194 - Introduction to Schrödinger Operators

Planned Outline

Part I

• Probability Measures

• Banach/Hilbert/Dual Spaces

Part II

• Recurrence

• Ergodic Theory

Part III

• Special Linear Matrices

• Spectral Theory

• SL2-cocycles and relations to Schrödinger Operators

Part IV

• Uniform Hyperbolicity

• Non-Uniform Hyperbolicity

Meeting Time: Wednesday from 2-3pm at Skye 268

Week 1 (Apr 3) - First meeting was over zoom. No material was covered.

Week 2 (Apr 10) - We discussed Part I and I also introduced the end goals of the reading.

Week 3 (Apr 17)  - We discussed Mean and Birkhoff Ergodic Theorems to gain an intuition on what the statement says.

Week 4 (Apr 24) - We discussed SL2, SVD, and stable and unstable subspaces.

Week 5 (May 1) - Discussed SL2-cocycles and the relation to  Schrödinger Operators 

Week 6 (May 8) - Uniform Hyperbolicity

Week 7 (May 15) - Took today off to talk about grad schools vs industry.

Week 8 (May 20) - Covered Johnson's Theorem (one direction)

Remaining weeks - Review previous material and work on presentation slides

Presentation Slides: Created by Darius Mahamedi, Tomas Sbardelotto Dos Santos, Alejandro Levya

References

• One-Dimensional Ergodic Schrödinger Operators by David Damanik and Jake Fillman

• Ergodic Theory by Einsiedler and Ward.

• Real Analysis by Folland 

• Uniform Positivity of the Lyapunov Exponent for Monotone Potentials Generated by the Doubling Map by Zhenghe Zhang