## Seminar (Functional Analysis): Random and Almost Periodic Schrödinger Operators

Wintersemester 2013/14

Time and Place

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PJSE 2 std. | Do 11:30-13:00 | SR10 | 3.10 |

Topics

Random and almost periodic Schrödinger operators serve in solid state physics as
models of disordered systems such as alloys, glasses and amorphous materials in the so
called tight binding approximation. In quantum mechanics, crystals are modeled by
Schrödinger operators with periodic potentials. It is an important problem to understand
what happens to a crystal if random impurities are introduced. Indeed, in a more realistic
setting such crystals will contain impurities, the precise locations of which are in
general unknown. All one might know is that these impurities occur with a certain
probability. During the last three decades, random Schrödinger operators became an
extremely extensive topic and much effort has been put to understand the properties of
random and almost periodic operators. Our main aim is to shed some light on a circle of
ideas and problems and to the mathematical machinery that has recently been built to
investigate it.

Presentations

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Title: |
Speaker: |
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10.10 | Introduction: Why random Schrödinger operators | Gerald Teschl | [Ki] |

24.10 | Discrete Schrödinger operators and spectral calculus | Markus Holzleitner | [Ki] |

31.10 | Random potentials | Tobias Schubhart | [Ki] |

07.11 | Ergodic operators | Damir Ferizović | [Ki] |

14.11 | The density of states: Definitions and existence | Franz Berger | [Ki] |

21.11 | The density of states: The geometric resolvent equation | Melanie Graf | [Ki] |

28.11 | The Wegner estimate | Oliver Skocek | [Ki] |

12.12 | Lifshitz tails | Noema Nicolussi | [Ki] |

23.01 | The spectrum and its physical interpretation | Zouhair Hadded | [Ki] |

References:

- J. Bourgain, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Princeton UP, 2005.
- H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, 2nd ed., Springer, 2008.
- W. Kirsch, An invitation to random Schrödinger operators, in: Panor. Syntheses, 25, Random Schrödinger operators, 1-119, 2008.
- A. Klein, Multiscale analysis and localization of random operators, in: Panor. Syntheses, 25, Random Schrödinger operators, 121-159, 2008.
- G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, AMS, 2000.

Course assessment

Preparation and presentation of a chosen topic.

Audience

Majors in Mathematics (master program, code MANS), Physics, ...

Auf Ihr Kommen freuen sich Aleksey Kostenko und Gerald Teschl