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Chapter 1．Discrete Spectrum 1

1．Introduction 1

2．States with small binding energy 8

3．Point interaction and its correspondence to boundary conditions 14

4．Particle in field of several point potentials 17

5．The Coulomb potential 20

6．Three-dimensional oscillator 28

7．Virial theorem and its generalizations 41

8．Identical particles and statistical physics 44

Chapter 2．Continuous Spectrum 49

1．Introduction.Wave functions of continuous spectrum with l=0 49

2．Motion with orbital angular momentum l≠0. Motion in the Coulomb field 57

3．Wave functions of continuous spectrum.Scattering cross-section 64

4．Optical theorem and its generalization 70

Chapter 3．Analytic Properties of Wave Function 75

1．Analytic properties of S-matrix 75

2．"Redundant"poles 84

3．Properties of residues of Sl(k) 88

4．Dispersion relations 93

Chapter 4．Inverse Scattering Problem 101

1．The Marchenko equation 101

2．Reflectionless potentials 106

3．Isospectral deformations of quantum oscillator 109

4．Isospectral deformations of the Schr？dinger equation and invariants of the Korteweg-de Vries equation 114

Chapter 5．The Green Functions and Perturbation Theory 123

1．Introduction.The Green function of the radial Schr？dinger equation 123

2．Regular method of obtaining of the Green functions 128

3．Some properties of the Green functions 134

4．The Green function for several free particles 136

5．Perturbation theory.Coordinate representation 138

6．Momentum representation 143

7．The Green function in momentum representation.Operator algebra 149

8．Scattering operator 154

9．Formulae for point potentials 159

10．Perturbation theory for continious spectrum 163

11．Convergence of series of perturbation theory 168

12．The time-dependent Green function 174

Chapter 6．Quasi-classical Approximation 177

1．Wave function in quasi-classical approximation 177

2．Quasi-classical approximation for the degenerate Fermi gas 182

3．Multi-dimensional case 188

4．Non-stationary problems 199

Chapter 7．Exact Solutions of Non-stationary Problems for Oscillator 213

1．Introduction 213

2．Wave function of oscillator with variable frequency under action of external force 214

3．Quantum oscillator under action of external force.Transition probabilities 218

4．Parametric excitation of quantum oscillator 220

5．Oscillator with variable frequency under action of external force.Transition probabilities 227

6．Quantum oscillator and adiabatic invariants 231

7．Quasi-energy of system under action of periodic force 236

8．The Heisenberg representation and canonical transformations 241

Chapter 8．Quasi-stationary States 255

1．Introduction.The Gamov theory 255

2．Wave functions 262

3．Example of quasi-stationary state 267

4．Decay of quasi-stationary state 272

5．Radioactive decay law 278

6．Generalization of normalization.Perturbation theory for quasi-stationary states 282

7．Asymptotic behaviour of wave function at r→∞ and t→∞ 285

8．Creation of unstable particle 289

9．Transition from quasi-stationary to stationary states 295

10．Collision time 298

11．Types of long-lived states 301

Appendix A．Specific cases of the Schr？dinger equation spectrum 305

Appendix B．Quasi-classical properties of highly excited levels in the Coulomb field 317

Bibliography 321

Subject Index 333