Contents
1 Essentials of Non-relativistic Two-Body Scattering Theory . . . . . . . .
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Boundary Condition of the Scattering State: Basic Concepts . . . .
2
1.3 Moller Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.4 Resolvent Equation and Lippmann–Schwinger Equation . . . . . . .
4
1.5 The S-Matrix, T-Matrix and Differential Cross Section . . . . . . . .
8
1.6 The Unitarity of S-Matrix and Optical Theorem . . . . . . . . . . . . . 11
2 Scattering Theory of Three-Particle System . . . . . . . . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Three-Particle Hamiltonian, Channel Hamiltonian,
Matrix-Elements for the Green’s Functions and T-Matrix . . . . . . 14
2.3 Boundary Conditions and Moller Operators . . . . . . . . . . . . . . . . 16
2.4 Difficulties with the 3-Particle Lippmann–Schwinger
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Non-uniqueness of the Boundary Conditions . . . . . . . . . . 17
2.4.2 Disconnectedness of the Kernel . . . . . . . . . . . . . . . . . . . 18
2.5 The Faddeev Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Faddeev Equations for Scattering States . . . . . . . . . . . . . . . . . . . 21
2.7 Three-Body Problem with Separable Potentials . . . . . . . . . . . . . . 25
2.7.1 How the Problems of Disconnectedness and the
Uniqueness of Boundary Conditions Are Tackled
Here? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Efimov’s Universal Three-Body Effect . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Relation Between Total Cross Section and Scattering Length
for Two Particles at Low Energies . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Efimov Effect in Three-Boson System . . . . . . . . . . . . . . . . . . . . 32
3.4 Salient Feature of Efimov Effect . . . . . . . . . . . . . . . . . . . . . . . . 36
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