Random Operators
Course in the fall semester 2010/2011 at TU Chemnitz.Dates:
Thursdays, 17:15-18:45, Room 41/705Please send a mail if you want to attend but you cannot because of the time!!
Overview:
Random Operators are used as models for disordered systems in Mathematical Physics. At the same time, their study has widened the scope of operator and spectral theory. The introduction offered in this course will shed light on both aspects.
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Content:
1. Introduction: the Anderson model2. Random and ergodic operators
2.1 Random operators - the definition
2.2 Ergodic operators have deterministic spectrum
2.3 Spectral types of selfadjoint operators
2.4 Dynamical characterization of spectral subspaces
2.5 The discrete Laplacian
2.6 Tensor products
2.7 Direct integrals
3. Localization for the Andersion model
(... after the paper by Graf cited below)
References:
Anderson, P.W.: Absence of Diffusion in Certain Random Lattices, Phys. Reviews, 109 (5), 1958, 1492--1505 .pdf
Anderson, P.W.: Local Moments and Localized States. Nobel Lecture, 8 December 1977 .pdf
Carmona, Rene ; Lacroix, Jean . Spectral theory of random Schroedinger operators. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. xxvi+587 pp. ISBN: 0-8176-3486-X
Graf, Gian Michele: Anderson localization and the space-time characteristic of continuum states. J. Statist. Phys. 75 (1994), no. 1-2, 337--346, .pdf
Pastur, Leonid ; Figotin, Alexander . Spectra of random and almost-periodic operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 297. Springer-Verlag, Berlin, 1992. viii+587 pp. ISBN: 3-540-50622-5
Stollmann, P.: Hilbertraumtheorie, Skript einer Vorlesung vom Wintersemester 2005/2006. .pdf
Stollmann, Peter . Caught by disorder. Bound states in random media. Progress in Mathematical Physics, 20. Birkhaeuser Boston, Inc., Boston, MA, 2001. xviii+166 pp. ISBN: 0-8176-4210-2
Veselic, Ivan . Existence and regularity properties of the integrated density of states of random Schroedinger operators. Lecture Notes in Mathematics, 1917. Springer-Verlag, Berlin, 2008. x+142 pp. ISBN: 978-3-540-72689-0
... to be continued