Schrodinger Operators

Teschl G.

Table of contents :
Preface……Page 8
Part 1. Mathematical Foundations of Quantum Mechanics……Page 10
0.1. Borel measures……Page 12
0.2. Integration……Page 14
0.3. The decomposition of measures……Page 16
0.4. Banach spaces……Page 17
0.5. Lebesgue spaces……Page 19
1.1. Hilbert spaces……Page 22
1.2. Orthonormal bases……Page 25
1.3. The projection theorem and the Riesz lemma……Page 28
1.4. Orthogonal sums and tensor products……Page 29
2.1. Some quantum mechanics……Page 32
2.2. Self-adjoint operators……Page 35
2.3. Resolvents and spectra……Page 45
2.4. Orthogonal sums of operators……Page 49
2.5. Self-adjoint extensions……Page 51
3.1. The spectral theorem……Page 54
3.2. More on Borel measures……Page 63
3.3. Spectral types……Page 65
3.4. Appendix: The Herglotz theorem……Page 67
4.1. Integral formulas……Page 72
4.2. Commuting operators……Page 76
4.3. The min-max theorem……Page 78
4.4. Estimating eigenspaces……Page 80
4.5. Tensor products of operators……Page 81
5.1. The time evolution and Stone’s theorem……Page 84
5.2. The RAGE theorem……Page 87
Part 2. Schrödinger Operators……Page 92
6.1. The Fourier transform……Page 94
6.2. The free Schrödinger operator……Page 97
6.3. The time evolution in the free case……Page 99
6.4. The resolvent and Green’s function……Page 101
7.1. Position and momentum……Page 104
7.2. Angular momentum……Page 105
7.3. The harmonic oscillator……Page 108
8.1. Relatively bounded operators and the Kato-Rellich theorem……Page 110
8.2. More on compact operators……Page 112
8.3. Relatively compact operators and Weyl’s theorem……Page 114
8.4. One-particle Schrödinger operators……Page 118
8.5. Sturm-Liouville operators……Page 119
9.1. The hydrogen atom……Page 128
9.2. Angular momentum……Page 130
9.3. The spectrum of the hydrogen atom……Page 133
9.4. Atomic Schrödinger operators……Page 136
10.1. Abstract theory……Page 144
10.2. Incoming and outgoing states……Page 147
10.3. Schrödinger operators with short range potentials……Page 149
Bibliography……Page 153
Glossary of notations……Page 154
Index……Page 156