An Invitation to Random Schr¨odinger operators - FernUniversität in ...

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CONTENTS
1. Preface 5
2. Introduction: Why random Schrödinger operators 7
2.1. The setting of quantum mechanics 7
2.2. Random Potentials 7
2.3. The one body approximation 10
3. Setup: The Anderson model 13
3.1. Discrete Schrödinger operators 13
3.2. Spectral calculus 15
3.3. Some more functional analysis 18
3.4. Random potentials 19
4. Ergodicity properties 23
4.1. Ergodic stochastic processes 23
4.2. Ergodic operators 25
5. The density of states 29
5.1. Definition and existence 29
5.2. Boundary conditions 35
5.3. The geometric resolvent equation 40
5.4. An alternative approach to the density of states 41
5.5. The Wegner estimate 43
6. Lifshitz tails 51
6.1. Statement of the Result 51
6.2. Upper bound 52
6.3. Lower bound 56
7. The spectrum and its physical interpretation 61
7.1. Generalized Eigenfunctions and the spectrum 61
7.2. The measure theoretical decomposition of the spectrum 65
7.3. Physical meaning of the spectral decomposition 67
8. Anderson localization 75
8.1. What physicists know 75
8.2. What mathematicians prove 76
8.3. Localization results 77
8.4. Further Results 78
9. The Green’s function and the spectrum 81
9.1. Generalized eigenfunctions and the decay of the Green’s function 81
9.2. From multiscale analysis to absence of a.c. spectrum 83
9.3. The results of multiscale analysis 85
9.4. An iteration procedure 86
9.5. From multiscale analysis to pure point spectrum 89
10. Multiscale analysis 93
10.1. Strategy 93