Chapter 2
Scattering Theory of Three-Particle
System
2.1 Introduction
In this chapter, we present a brief and simple formulation of Faddeev’s three-particle
scattering theory. Recognizing the basic complexities involved while extending the
two-body Lippmann–Schwinger equation to the three-particle case with pair-wise
potentials, Faddeev [7] showed how the problems of disconnectedness of the scattering matrix in the three-body kernel can be resolved by expressing the three-particle
T-matrix as a sum of three terms where each term is expressed in terms of the other
two and the two body potentials are replaced by the corresponding T-matrices for
the two particles. He also successfully tackled the problem of non-uniqueness of
the boundary conditions for different channels encountered in the usual Lippmann–
Schwinger equation for three-particle scattering. Faddeev [7] undertook a detailed
investigation to study the analytic behavior of the scattering amplitude of the resulting
integral equations to show that these equations have completely continuous(compact)
kernel and has unique solutions at scattering energies. In the last sub-section, we
discuss the solution of three-particle system using Schrodinger equation with separable pair-wise potentials and derive the integral equation for the scattering amplitude
of a particle by the bound state of the other two. The idea is to demonstrate how the
problems of disconnectedness of the scattering matrix and the non-uniqueness of the
boundary condition are resolved in this approach.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_2
13
Scattering Theory of Three-Particle
System
2.1 Introduction
In this chapter, we present a brief and simple formulation of Faddeev’s three-particle
scattering theory. Recognizing the basic complexities involved while extending the
two-body Lippmann–Schwinger equation to the three-particle case with pair-wise
potentials, Faddeev [7] showed how the problems of disconnectedness of the scattering matrix in the three-body kernel can be resolved by expressing the three-particle
T-matrix as a sum of three terms where each term is expressed in terms of the other
two and the two body potentials are replaced by the corresponding T-matrices for
the two particles. He also successfully tackled the problem of non-uniqueness of
the boundary conditions for different channels encountered in the usual Lippmann–
Schwinger equation for three-particle scattering. Faddeev [7] undertook a detailed
investigation to study the analytic behavior of the scattering amplitude of the resulting
integral equations to show that these equations have completely continuous(compact)
kernel and has unique solutions at scattering energies. In the last sub-section, we
discuss the solution of three-particle system using Schrodinger equation with separable pair-wise potentials and derive the integral equation for the scattering amplitude
of a particle by the bound state of the other two. The idea is to demonstrate how the
problems of disconnectedness of the scattering matrix and the non-uniqueness of the
boundary condition are resolved in this approach.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_2
13