1. Random Operators ............................................. 1
1.1. Physical Background ..................................... 1
1.2. Model and Notation ...................................... 2
1.3. Transport Properties and Spectral Types ................. 6
1.4. Fluctuation Boundaries of the Spectrum ................. 10
2. Existence of the Integrated Density of States ............... 13
2.1. Schrödinger Operators on Manifolds: Motivation ......... 15
2.2. Random Schrödinger Operators on Manifolds:
Definitions ............................................ 17
2.3. Non-Randomness of Spectra and Existence of the IDS ..... 21
2.4. Measurability .......................................... 26
2.5. Bounds on the Heat Kernels Uniform in ω ................ 30
2.6. Laplace Transform and Ergodic Theorem .................. 37
2.7. Approach Using Dirichlet-Neumann Bracketing ............ 39
2.8. Independence of the Choice of Boundary Conditions ...... 42
3. Wegner Estimate ............................................. 45
3.1. Continuity of the IDS .................................. 46
3.2. Application to Anderson Localisation ................... 50
3.3. Resonances of Hamiltonians on Disjoint Regions ......... 53
4. Wegner's Original Idea. Rigorous Implementation ............. 57
4.1. Spectral Averaging of the Trace of the Spectral
Projection ............................................. 57
4.2. Improved Volume Estimates .............................. 61
4.3. Sparse Potentials ...................................... 64
4.4. Locally Continuous Coupling Constants .................. 66
4.5. Potentials with Small Support .......................... 69
4.6. Hölder Continuous Coupling Constants ................... 72
4.7. A Partial Integration Formula for Singular
Distributions .......................................... 73
4.8. Coupling Constants with Bernoulli Disorder ............. 74
4.9. Single Site Potentials with Changing Sign .............. 74
4.10.Uniform Wegner Estimates for Long Range Potentials ..... 75
5. Lipschitz Continuity of the IDS ............................. 79
5.1. Partition of the Trace into Local Contributions ........ 80
5.2. Spectral Averaging of Resolvents ....................... 83
5.3. Stone's Formula and Spectral Averaging of
Projections ............................................ 84
5.4. Completion of the Proof of Theorem 5.0.1 ............... 86
5.5. Single Site Potentials with Changing Sign .............. 87
5.6. The Finite Section Method for Multi-Level Laurent
Matrices ............................................... 95
5.7. Unbounded Coupling Constants and Magnetic Fields ....... 96
A. Properties of the Spectral Shift Function ................... 99
A.l. The SSF for Trace Class Perturbations .................. 99
A.2. The SSF for Schrödinger Operators and the
Invariance Principle .................................. 102
A.3. Singular Value Estimates .............................. 103
A.4. Bounds on the SSF for Schrцdinger Operators ........... 108
References .................................................... 113
Index ......................................................... 139
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