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Ⅺ：SCATTERING THEORY 1

1．An overview of scattering phenomena 1

2．Classical particle scattering 5

3．The basic principles of scattering in Hilbert space 16

Appendix 1 Stationary phase methods 37

Appendix 2 Trace ideal properties of f（x）g（-i？） 47

Appendix 3 A general invariance principle for wave operators 49

4．Quantum scattering Ⅰ：Two-body case 54

5．Quantum scattering Ⅱ：N-body case 75

6．Quantum scattering Ⅲ：Eigenfunction expansions 96

Appendix Introduction to eigenfunction expansions by the auxiliary space method 112

7．Quantum scattering Ⅳ：Dispersion relations 116

8．Quantum scattering Ⅴ：Central potentials 121

A．Reduction of the S-matrix by symmetries 121

B．The partial wave expansion and its convergence 127

C．Phase shifts and their connection to the Schr？dinger equation 129

D．The variable phase equation 133

E．Jost functions and Levinson's theorem 136

F．Analyticity of the partial wave amplitude for generalized Yukawa potentials 143

G．The Kohn variational principle 147

Appendix 1 Legendre polynomials and spherical Bessel functions 149

Appendix 2 Jost solutions for oscillatory potentials 155

Appendix 3 Jost solutions and the fundamental problems of scattering theory 164

9．Long-range potentials 169

10．Optical and acoustical scattering Ⅰ：Schr？dinger operator 184

Appendix Trace class properties of Green's functions 203

11．Optical and acoustical scattering Ⅱ：The Lax-Phillips method 210

Appendix The twisting trick 241

12．The linear Boltzmann equation 243

13．Nonlinear wave equations 252

Appendix Conserved currents 278

14．Spin wave scattering 285

15．Quantum field scattering Ⅰ：The external field 293

16．Quantum field scattering Ⅱ：The Haag-Ruelle theory 317

17．Phase space analysis of scattering and spectral theory 331

Appendix The RAGE theorem 340

Notes 344

Notes on scattering theory on C-algebras 382

Problems 385

MATERIAL PREPRINTED FROM VOLUME Ⅳ 406

ⅩⅢ.6 The absence of singular continuous spectrum Ⅰ：General theory 406

ⅩⅢ.7 The absence of singular continuous spectrum Ⅱ：Smooth perturbations 411

A．Weakly coupled quantum systems 421

B．Positive commutators and repulsive potentials 427

C．Local smoothness and wave operators for repulsive potentials 433

ⅩⅢ.8 The absence of singular continuous spectrum Ⅲ：Weighted L2 spaces 438

Notes 447

Problems 450

List of Symbols 455

Index 457