Schroedinger operators and mathematical methods in QM

Teschl G.

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 26
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 39
Part 1. Mathematical Foundations of Quantum Mechanics……Page 42
1.1. Hilbert spaces……Page 44
1.2. Orthonormal bases……Page 46
1.3. The projection theorem and the Riesz lemma……Page 50
1.4. Orthogonal sums and tensor products……Page 51
1.5. The C* algebra of bounded linear operators……Page 53
1.6. Weak and strong convergence……Page 55
1.7. Appendix: The Stone–Weierstraß theorem……Page 57
2.1. Some quantum mechanics……Page 60
2.2. Self-adjoint operators……Page 63
2.3. Resolvents and spectra……Page 75
2.4. Orthogonal sums of operators……Page 80
2.5. Self-adjoint extensions……Page 82
2.6. Appendix: Absolutely continuous functions……Page 85
3.1. The spectral theorem……Page 88
3.2. More on Borel measures……Page 99
3.3. Spectral types……Page 102
3.4. Appendix: The Herglotz theorem……Page 104
4.1. Integral formulas……Page 110
4.2. Commuting operators……Page 113
4.3. The min-max theorem……Page 116
4.4. Estimating eigenspaces……Page 117
4.5. Tensor products of operators……Page 118
5.1. The time evolution and Stone’s theorem……Page 120
5.2. The RAGE theorem……Page 123
5.3. The Trotter product formula……Page 128
6.1. Relatively bounded operators and the Kato–Rellich theorem……Page 130
6.2. More on compact operators……Page 132
6.3. Hilbert–Schmidt and trace class operators……Page 135
6.4. Relatively compact operators and Weyl’s theorem……Page 141
6.5. Strong and norm resolvent convergence……Page 145
Part 2. Schrödinger Operators……Page 150
7.1. The Fourier transform……Page 152
7.2. The free Schrödinger operator……Page 157
7.3. The time evolution in the free case……Page 159
7.4. The resolvent and Green’s function……Page 160
8.1. Position and momentum……Page 164
8.2. Angular momentum……Page 166
8.3. The harmonic oscillator……Page 169
9.1. Sturm-Liouville operators……Page 172
9.2. Weyl’s limit circle, limit point alternative……Page 176
9.3. Spectral transformations……Page 183
10.1. Self-adjointness and spectrum……Page 192
10.2. The hydrogen atom……Page 193
10.3. Angular momentum……Page 196
10.4. The eigenvalues of the hydrogen atom……Page 199
10.5. Nondegeneracy of the ground state……Page 201
11.1. Self-adjointness……Page 204
11.2. The HVZ theorem……Page 206
12.1. Abstract theory……Page 212
12.2. Incoming and outgoing states……Page 215
12.3. Schrödinger operators with short range potentials……Page 217
Part 3. Appendix……Page 222
A.1. Borel measures in a nut shell……Page 224
A.2. Extending a premasure to a measure……Page 228
A.3. Measurable functions……Page 233
A.4. The Lebesgue integral……Page 235
A.5. Product measures……Page 240
A.6. Decomposition of measures……Page 242
A.7. Derivatives of measures……Page 245
Bibliography……Page 248
Glossary of notations……Page 250
Index……Page 252