# Spectral Theory of Schrödinger Operators

Share this page *Edited by *
*Rafael del Río; Carlos Villegas-Blas*

This volume gathers the articles based on a series of lectures from a
workshop held at the Institute of Applied Mathematics of the National
University of Mexico. The aim of the book is to present to a non-specialized
audience the basic tools needed to understand and appreciate new trends of
research on Schrödinger operator theory.

Topics discussed include various aspects of the spectral theory of
differential operators, the theory of self-adjoint operators, finite rank
perturbations, spectral properties of random Schrödinger operators, and
scattering theory for Schrödinger operators.

The material is suitable for graduate students and research mathematicians
interested in differential operators, in particular, spectral theory of
Schrödinger operators.

This book is published in cooperation with Sociedad Matemática Mexicana

#### Readership

Graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrödinger operators.

# Table of Contents

## Spectral Theory of Schrodinger Operators

- Contents v6 free
- Preface vii8 free
- Topics from spectral theory of differential operators 110 free
- Spectral theory for self-adjoint extensions 5160
- 1. Introduction 5160
- 2. General notation and results 5665
- 3. The spectral calculus 6069
- 4. Spectral subspaces 6372
- 5. Schatten classes 6675
- 6. Spectral decomposition and dynamics 6776
- 7. Self – adjoint extensions 7281
- 8. Spectra inside one gap 7786
- 9. Absolutely continuous spectra 9099
- 10. Coupling between gaps 93102
- References 95104

- Integrated density of states and Wegner estimates for random Schrödinger operators 97106
- 1. Random operators 98107
- 2. Existence of the integrated density of states 105114
- 2.1. Schrödinger operators on manifolds: motivation 107116
- 2.2. Random Schrodinger operators on manifolds: definitions 108117
- 2.3. Non-randomness of spectra and existence of the IDS 112121
- 2.4. Measurability 116125
- 2.5. Bounds on the heat kernels uniform in ω 119128
- 2.6. Laplace transforms and Ergodic Theorem 125134
- 2.7. Approach using Dirichlet-Neumann bracketing 127136
- 2.8. Independence of the choice of boundary conditions 129138

- 3. Wegner estimate 129138
- 4. Wegner's original idea. Rigorous implementation 139148
- 4.1. Spectral averaging of the trace of the spectral projection 139148
- 4.2. Improved volume estimate 142151
- 4.3. Sparse potentials 144153
- 4.4. Locally continuous coupling constants 145154
- 4.5. Potentials with small support 147156
- 4.6. Hölder continuous coupling constants 150159
- 4.7. Single site potentials with changing sign 150159
- 4.8. Uniform Wegner estimates for long range potentials 151160

- 5. Lipschitz continuity of the IDS 152161
- 5.1. Partition of the trace into local contributions 153162
- 5.2. Spectral averaging of resolvents 155164
- 5.3. Stone's formula and spectral averaging of projections 156165
- 5.4. Completion of the proof of Theorem 5.0.1 158167
- 5.5. Single site potentials with changing sign 158167
- 5.6. Unbounded coupling constants and magnetic fields 162171

- Appendix A. Properties of the spectral shift function 163172
- References 167176
- Index 181190 free

- Singular and supersingular perturbations: Hilbert space methods 185194
- 1. Introduction 185194
- 2. Regular rank one perturbations and the first resolvent formula 189198
- 3. Singular perturbations and the second resolvent formula 190199
- 4. Supersingular perturbations I: H–3-case 195204
- 5. Supersingular perturbations II: general case 201210
- 6. Point interaction in R3 with p-symmetric eigenfunctions 209218
- References 214223

- Scattering and spectral properties of two surface models 217226