Math 521: Advanced Topics in Real Analysis - Ergodic Schrödinger Operators
We will present the general theory of ergodic Schrödinger operators, including almost sure invariance of spectra and spectral types, the definition and general properties of the integrated density of states and the Lyapunov exponent, and Kotani theory. Almost-periodic and random operators will be studied in detail. Operators generated by other underlying dynamics, such as the skew-shift and the doubling map, will also be discussed. There are no required prerequisites, though familiarity with linear algebra, probability theory, basic real and complex analysis, and some functional analysis will be helpful.
TuTh 9:25 AM -- 10:40 AM
Herman Brown 423
[my last name] at rice dot edu
Herman Brown 434
Th 2:15 PM -- 3:30 PM or by appointment
Green's Function Estimates for Lattice Schrödinger Operators and Applications
, Princeton University Press, Princeton, 2005
R. Carmona and J. Lacroix,
Spectral Theory of Random Schrödinger Operators
, Birkhäuser, Boston, 1990
H. Cycon, R. Froese, W. Kirsch and B. Simon,
Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
, Springer, Berlin, 1987
L. Pastur and A. Figotin,
Spectra of Random and Almost-Periodic Operators
, Springer, Berlin, 1992
Presentation of a Research Paper
There will be regular homework assignments. All homework should be turned in during class on or before the due date. The homework is not pledged. You are encouraged to discuss the homework and to work together on the problems, though you should write up your own submission.
Each student will be asked to give a public presentation of a fundamental research paper in the field. The instructor will recommend a number of papers, but other suggestions are welcome.