Spectral Theory of Random Schrödinger Operators
Lecturer:
Prof. Dr. Simone Warzel
Exercises:
Michael Fauser
Time and Location
Lecture: M 16:00-17:30, Th 8:30-10:00, MI 03.08.011
Exercises: M 14:15-15:45, MI 03.06.011
Office hours of Michael Fauser: W 15:00-17:00, MI 03.06.021
Summary
This course offers an introduction to the theory of random Schrödinger operators. Such operators describe effects of randomness on the spectra and dynamics of disordered quantum systems.
Among the physically interesting effects are Anderson localization and the Quanten Hall Effect.
A mathemematical description requires elements of spectral theory on Hilbert spaces which range beyond the narrow interest of random operators. Examples are the relation of spectra and dynamics, ergodic theorems,
elements of harmonic analysis and topological concepts in operator theory.
Prerequisite is some familiarity with the theory of Hilbert spaces. The first part of the course will be an introduction to the spectral theory of (bounded) self-adjoint operators.
Lecture Notes
- VLWoche1.pdf
- VLWoche2.pdf
- Chapter2.pdf Material containing Wiener and RAGE theorem (Ch. 2.2) - the other material in this chapter is not relevant for this course.
- Chapter3.pdf Ergodic operators were only discussed for the d-dimensional lattice - not more general graphs.
- Chapter4.pdf Section 4.5 was omitted in the course.
- Chapter5.pdf In Section 5.3-4 we only discussed Theorem 5.4 and Corollary 5.9. The rest (including Section 5.5) was omitted in the course.
- Chapter6.pdf Relevant for this course are Section 6.1-3. In Section 6.3 we only discussed Lemma 6.7 and Theorem 6.8. The rest (including Section 6.4 and Subsection 6.3.1.) was omitted in the course.
- Chapter7_Part1.pdf So far, we have discussed Sec. 7.1.1. and Theorem 8.1 from Chapter8.pdf
- Combes_Thomas.pdf
- SchattenClasses.pdf Some of these facts are just collected and will be used without proof. (See references).
- IQHE.pdf Material related to the Integer Quantum Hall Effect.
- ChapterKubo.pdf Contains the material on the decay of the Fermi projection in the localization regime (Theorem 11.6). The outlook in the last lecture on the derivation of the Kubo formula is also contained here.
- References.pdf
Exercise sheets
- Exercise sheet 1, solutions
- Exercise sheet 2, solutions
- Exercise sheet 3, solutions
- Exercise sheet 4, solutions
- Exercise sheet 5, solutions
- Exercise sheet 6, solutions
- Exercise sheet 7, solutions
- Exercise sheet 8, solutions
- Exercise sheet 9, solutions
- Exercise sheet 10, solutions
- Exercise sheet 11, solutions
- Exercise sheet 12, solutions
Annoucements
- 28.07.2014: Klausurtermin im Seminarraum 03.08.011. Please reply to the Email dated 8.7. concerning the exam modus
- 10.07.2014: A small change was made to Exercise 11.1(d) (concerning the constants C and mu).
- 25.06.2014: In Exercise 11.1(b), an additional assumption was added.
- 17.06.2014: In Exercise 9.1(ii), s-moment regularity needs to be assumed for all s in order to be applied in (iii). The exercise sheet was changed accordingly.
- 12.06.2014: The wording of Exercise 6.2 was changed so as to be more precise.
- 12.06.2014: There will be no lecture on Thursday, 12.06. Instead, there is an extra Exercise class from 8:30-10:00 on that day (in MI 03.08.011).
- 20.05.2014: Please note that the definition of the discrete Dirichlet Laplacian in Exercise 5.3 was changed. In the previous version of the exercise sheet, the last inequality in Exercise 5.3 was not correct.
- 22.04.2014: There was a mistake in Exercise 2.2(iii): The claim should be an inequality instead of an equality.
- 22.04.2014: There was a mistake in the definition of the Fourier transform in Exercise 2.4. The integral should be with respect to the variable E instead of t.
- 15.04.2014: The office hours (and ersatz exercise class) will take place every Wednesday, 15:00-17:00, in MI 03.08.022B.
- 10.04.2014: Please participate in the poll to find a date for the office hours of Michael Fauser. The office hours are particularly intended for those of you who are not able to attend the exercise class. Depending on your wishes, we can discuss the latest exercises and/or other course-related questions.
Additional literature
- R. Carmona and J.Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, MA, 1990
- W. Kirsch, An invitation to random Schrödinger operators
- W. Kirsch, Random Schrödinger operators: a course, pp. 264–370 in H. Holden and A. Jensen (Eds.), Schrödinger operators, Lecture Notes in Physics 345, Springer, Berlin, 1989
- L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin, 1992
- G. Teschl, Mathematical methods in quantum mechanics, AMS 2009