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Journal of Mathematical Physics

Exact Solution of the Faddeev Equations for a One-Dimensional System

L. R. Dodd

Published Jan 1, 1970• 50 citations
Publisher

Abstract

The Faddeev equations for the one-dimensional system consisting of three identical particles interacting through δ-function potentials are shown to be exactly solvable. The scattering solutions in the form of half-off-shell rearrangement and elastic amplitudes are constructed explicitly and the ground state of the system is derived.
DOI
10.1063/1.1665049

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The exact solution to the one-dimensional Schr\"odinger equation which describes the motion of a particle scattering off a finite set of $\ensuremath{\delta}$-function potentials of arbitrary strengths and positions is found. The behavior of the TTT matrix both on and off the energy shell is also precisely given. Localization of states is discussed.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.
The reformulation of the one-dimensional three-body problems with boundary condition and delta function interactions, based on the Sommerfeld–Maluzhinetz transformation, is presented. The argumentation is carried out as exemplified by two models–the exactly soluble model of two identical particles interacting through delta potential and each of which interacts with a third one through boundary condition interactions, and a model of two identical particles and a fixed wall, all interactions being of the delta function type. The problems are reduced to those of solving coupled systems of functional equations for the Sommerfeld transforms of the wavefunction. The functional properties of the transforms are then used to derive expressions relating them to the half off-shell extensions of the elastic and exchange probability amplitudes as defined in the Faddeev–Lovelace approach.
We derive some new equations for the wavefunction for the three-body scattering problem using the ``partial-wavefunction approach.'' All scattering processes are solutions of inhomogeneous coupled integral equations for the scattered parts of the total wavefunction. We show how the Pauli principle is incorporated when some or all of the particles are identical. This leads to a reduction in the number of independent scattered partial wavefunctions when some of the particles are identical. The kernels in the integral equations contain no δ functions but each scattered partial wavefunction does have a self-coupling term. We apply the method to the scattering of three identical bosons interacting through zero-range pair potentials. The method gives the same equation for this problem as obtained by the Faddeev T-matrix approach.