Ma 191c - Spring 2026

Ma 191c: Selected Topics in Mathematics - Spectral Theory and Anderson Localization (Spring 2026)


Instructor Lingfu Zhang (lingfuz@caltech Office Linde Hall 358)

Class time 10:30-11:50 on Tuesdays and Thursdays
Class location Linde Hall 255
Office hour 14:00 - 15:00 on Tuesdays

Course description


This topic course provides an introduction to random operator theory, with a focus on the mathematical theory of localization and delocalization phenomena for discrete random Schrödinger operators (i.e., the Anderson model; see Anderson localization). We may also discuss related random matrices and localization on general graphs.

This course is intended for advanced undergraduate students and PhD students from PMA and related departments.

Grading Based on participation and final presentation.

Reference material

Main textbook:
  • [AW] Random Operators: Disorder Effects on Quantum Spectra and Dynamics by Michael Aizenman and Simone Warzel.

    • Other useful books:
  • [Sto] Caught by Disorder: Bound States in Random Media by Peter Stollmann.
  • [Kir] An Invitation to Random Schrödinger Operators by Werner Kirsch.

    • A list of papers (research or survey):
  • [AM] Aizenman, Michael and Molchanov, Stanislav. Localization at large disorder and at extreme energies: an elementary derivation.
  • [FS] Fröhlich, Jürg and Spencer, Thomas. Absence of diffusion in the Anderson tight binding model for large disorder or low energy.
  • [Li] Li, Linjun. On the Manhattan pinball problem.
  • [Spe] Spencer, Thomas. Duality, statistical mechanics and random matrices.
  • More to be added.




    • Schedule

    March 31 Introduction, background/motivation from physics, Manhattan pinball / random mirror models, a brief history. (Chapter 1 of [AW]; see also [Spe], [Li])
    Apr 2 Resolvent and spectrum, Weyl's criterion, spectrum of random Schrödinger operator, decomposition (Section 2.1, part of Section 3.2, and Appendix A of [AW])
    (Notes for Background, pinball problem, and basics of spectrum.)
    Apr 7 Spectral measure and spectral theorem, connection to dynamics, RAGE theorem. (Section 2.2, 2.3, 2.4 of [AW])
    (Short notes on RAGE theorem and on the projection operator.)
    Apr 9 General ergodic operator, Pastur's theorem. (Section 3.1, 3.2 of [AW])
    Apr 14 Almost Mathieu operator, density of state, Green function. (Section 3.1, 3.3, 3.4 of [AW])
    (Short notes on ergodic operators.)
    Apr 16 Finite dimensional pertubation, Simon-Wolff criterion. (Section 5.1, 5.3 of [AW])
    Apr 21 Spectral averaging, Wegner estimate, zero-one law boost of Simon-Wolff. (Section 5.2, 5.6 of [AW])
    (Notes for Green functions, and perturbation.)
    Apr 23
    Apr 28
    Apr 30
    May 5
    May 7
    May 12
    May 14
    May 19
    May 21
    May 26
    May 28