Mathematical Methods in Quantum Mechanics

G. Teschl

Table of contents :
Preface……Page 8
Part 0. Preliminaries……Page 12
0.1. Warm up: Metric and topological spaces……Page 14
0.2. The Banach space of continuous functions……Page 21
0.3. The geometry of Hilbert spaces……Page 25
0.4. Completeness……Page 30
0.5. Bounded operators……Page 31
0.6. Lebesgue Lp spaces……Page 33
0.7. Appendix: The uniform boundedness principle……Page 38
Part 1. Mathematical Foundations of Quantum Mechanics……Page 40
1.1. Hilbert spaces……Page 42
1.2. Orthonormal bases……Page 44
1.3. The projection theorem and the Riesz lemma……Page 47
1.4. Orthogonal sums and tensor products……Page 49
1.5. The C* algebra of bounded linear operators……Page 51
1.6. Weak and strong convergence……Page 52
1.7. Appendix: The Stone–Weierstraß theorem……Page 55
2.1. Some quantum mechanics……Page 58
2.2. Self-adjoint operators……Page 61
2.3. Resolvents and spectra……Page 72
2.4. Orthogonal sums of operators……Page 78
2.5. Self-adjoint extensions……Page 79
2.6. Appendix: Absolutely continuous functions……Page 83
3.1. The spectral theorem……Page 86
3.2. More on Borel measures……Page 96
3.3. Spectral types……Page 100
3.4. Appendix: The Herglotz theorem……Page 102
4.1. Integral formulas……Page 108
4.2. Commuting operators……Page 111
4.3. The min-max theorem……Page 114
4.4. Estimating eigenspaces……Page 115
4.5. Tensor products of operators……Page 116
5.1. The time evolution and Stone’s theorem……Page 118
5.2. The RAGE theorem……Page 121
5.3. The Trotter product formula……Page 126
6.1. Relatively bounded operators and the Kato–Rellich theorem……Page 128
6.2. More on compact operators……Page 130
6.3. Hilbert–Schmidt and trace class operators……Page 133
6.4. Relatively compact operators and Weyl’s theorem……Page 139
6.5. Strong and norm resolvent convergence……Page 142
Part 2. Schrödinger Operators……Page 148
7.1. The Fourier transform……Page 150
7.2. The free Schrödinger operator……Page 153
7.3. The time evolution in the free case……Page 155
7.4. The resolvent and Green’s function……Page 156
8.1. Position and momentum……Page 160
8.2. Angular momentum……Page 162
8.3. The harmonic oscillator……Page 165
9.1. Sturm-Liouville operators……Page 168
9.2. Weyl’s limit circle, limit point alternative……Page 172
9.3. Spectral transformations……Page 179
10.1. Self-adjointness and spectrum……Page 188
10.2. The hydrogen atom……Page 189
10.3. Angular momentum……Page 192
10.4. The eigenvalues of the hydrogen atom……Page 195
10.5. Nondegeneracy of the ground state……Page 197
11.1. Self-adjointness……Page 200
11.2. The HVZ theorem……Page 202
12.1. Abstract theory……Page 208
12.2. Incoming and outgoing states……Page 211
12.3. Schrödinger operators with short range potentials……Page 213
Part 3. Appendix……Page 218
A.1. Borel measures in a nut shell……Page 220
A.2. Extending a premasure to a measure……Page 224
A.3. Measurable functions……Page 229
A.4. The Lebesgue integral……Page 231
A.5. Product measures……Page 235
A.6. Decomposition of measures……Page 238
A.7. Derivatives of measures……Page 240
Bibliography……Page 244
Glossary of notations……Page 246
Index……Page 248