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 presentations.

Reference material

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

    • Other useful materials:
  • [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.
  • [Kir] Kirsch, Werner. An Invitation to Random Schrödinger Operators.
  • [Spe] Spencer, Thomas. Duality, statistical mechanics and random matrices.
  • [Sto] Stollmann, Peter. Caught by Disorder: Bound States in Random Media.

    • A list of papers for final presentations:
  • [AW1] Aizenman, Michael and Warzel, Simone. Localization Bounds for Multiparticle Systems.
  • [Bou1] Bourgain, Jean. On Localization for Lattice Schrödinger Operators Involving Bernoulli Variables.
  • [Bou2] Bourgain, Jean. A lower bound for the Lyapounov exponents of the random Schrödinger.
  • [BSV] Bordenave, Charles and Sen, Arnab and Virág, Bálint. Mean quantum percolation.
  • [CCFST] J T Chayes, L Chayes, J R Franz, J P Sethna and S A Trugman On the density of state for the quantum percolation problemr.
  • [DS] Drogin, Reuben and Smart, Charles. The regular tree Anderson model at low disorder.
  • [JZ] Jitomirskaya, Svetlana and Zhu, Xiaowen. Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model.
  • [Li1] Li, Linjun. On the Manhattan pinball problem.
  • [Li2] Li, Linjun. Polynomial bound for the localization length of Lorentz mirror model on the 1D cylinder.
  • [Sch] Schenker, Jeffrey. Eigenvector Localization for Random Band Matrices with Power Law Band Width.




    • 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 Simplicity of Spectrum, path expansion. (Section 5.4, 6.1, 6.2 of [AW])
    (Notes for path expansion.)
    Apr 28 Fractional moment bound, eigenfunction correlator, strong dynamical localization. (Section 6.3, 7.1, 7.2 of [AW])
    May 5 Bounds for eigenfunction correlator, DOS near edge. (Section 4.4, 7.3 of [AW])
    (Notes for eigenfunction correlator and localization.)
    May 7 Lifshitz tail, Combes-Thomas bound. (Section 4.3, 4.4, 4.5, 10.3 of [AW])
    (Notes for Lifshitz tail and Combes-Thomas bound.)
    May 12 Induction step of multi-scale analysis. (Section 3.1, 3.2 or [Sto], Section 10 of [Kir])
    May 14 From multi-scale analysis to localization, generalized eigenfunctions. (Section 9 of [Kir])
    (Notes for multi-scale analysis.)
    May 19 Spectrum via generalized eigenvalues, phase diagram on infinite regular trees. (Section 7.1 of [Kir], Section 16.1, 16.2, 16.3 of [AW])
    May 21 Localization and delocalization on infinite regular trees. (Section 15.2, 16.4, 16.5 of [AW])
    (Notes for regular trees.)
    May 26 Final presentations.
    May 28 Final presentations.