Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

On the application of the Sommerfeld–Maluzhinetz transformation to some one-dimensional three-particle problems

K. Lipszyc

Published May 1, 1980• 10 citations
Publisher

Abstract

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.

DOI
10.1063/1.524558

Related papers

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

L. R. Dodd

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.

Journal of Mathematical Physics•Jan 1, 1970•50 citations

Integral representation of one-dimensional three particle scattering for δ function interactions

Alejandro Amaya-Tapia, G. Gasaneo, S. Yu. Ovchinnikov +2 more

The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.

arXiv:math-ph/0406019•Journal of Mathematical Physics•Aug 6, 2004•7 citations

Study of Exactly Soluble One-Dimensional N-Body Problems

James McGuire

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.

Journal of Mathematical Physics•May 1, 1964•745 citations

Exact solution of the bound-state Faddeev-Yakubovsky equations for one-dimensional systems with a δ-function interaction

Alexander L. Zubarev, Victor B. Mandelzweig

The Faddeev-Yakubovsky equations for four identical spinless particles in one-dimensional space interacting via δ-function potentials are solved analytically for a bound state.Received 13 February 1995DOI:https://doi.org/10.1103/PhysRevC.52.509©1995 American Physical Society

Physical Review C•Aug 1, 1995•2 citations

An exactly soluble three-body problem in one dimension

Christof Jung

An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.

Canadian Journal of Physics•Jun 1, 1980•29 citations

Approach to the Three-Body Scattering Problem

John R. Jasperse

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.

Journal of Mathematical Physics•Nov 1, 1968•1 citation
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.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.
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.
An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.
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.