© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Mathematical Physics

Study of Exactly Soluble One-Dimensional N-Body Problems

James McGuire

Published May 1, 1964• 745 citations

Abstract

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.

DOI: 10.1063/1.1704156

Related papers

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

Solution of a Three-Body Scattering Problem in One Dimension

Carlo Marchioro

The scattering problem of three equal particles interacting via repulsive inverse-cube forces is solved in one dimension both in the classical and the quantal cases. The quantal S matrix is similar to that produced by repulsive δ-function interaction of infinite strength.

Journal of Mathematical Physics•Jul 1, 1970•48 citations

Three interacting particles in one dimension: An algebraic approach

James McGuire, C. A. Hurst

An algebraic formulation of the problem of three particles in one dimension is given, where the particles interact with delta function potentials of arbitrary strength and have almost arbitrary mass. An algebraic formulation is taken to mean that the steps implied from formulation to solution involve finite algebra. The canonical example is equal mass particles interacting with equal strength delta function potentials, where the Bethe ansatz holds and the solution involves only sums of products of matrices with elements that are rational functions of a complex variable. When the Bethe ansatz fails the Sommerfeld diffraction ansatz is satisfied if a condition of internal consistency is met. This condition of internal consistency requires the solution to a Riemann–Hilbert functional equation with an algebraic coefficient. The solution to this functional equation is an analytic, but not generally a meromorphic function. It is demonstrated that an asymptotic solution may be constructed within the domain of algebraic functions.

Journal of Mathematical Physics•Jan 1, 1988•14 citations

S Matrix for the One-Dimensional N-Body Problem with Repulsive or Attractive δ-Function Interaction

Chen Ning Yang

For $N$ particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the $S$ matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.

Advanced series in mathematical physics•Mar 1, 1990•0 citations

S Matrix for the One-Dimensional N -Body Problem with Repulsive or Attractive δ -Function Interaction

C. N. Yang

For $N$ particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the $S$ matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.

Physical Review•Apr 25, 1968•361 citations

Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials

Francesco Calogero, Carlo Marchioro

The exact solution is presented of the scattering problem of three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.

Journal of Mathematical Physics•Sep 1, 1974•101 citations

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.

An algebraic formulation of the problem of three particles in one dimension is given, where the particles interact with delta function potentials of arbitrary strength and have almost arbitrary mass. An algebraic formulation is taken to mean that the steps implied from formulation to solution involve finite algebra. The canonical example is equal mass particles interacting with equal strength delta function potentials, where the Bethe ansatz holds and the solution involves only sums of products of matrices with elements that are rational functions of a complex variable. When the Bethe ansatz fails the Sommerfeld diffraction ansatz is satisfied if a condition of internal consistency is met. This condition of internal consistency requires the solution to a Riemann–Hilbert functional equation with an algebraic coefficient. The solution to this functional equation is an analytic, but not generally a meromorphic function. It is demonstrated that an asymptotic solution may be constructed within the domain of algebraic functions.

For

N

particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the

S

matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.

For

N

particles with equal mass, interacting with repulsive or attractive $\ensuremath{\delta}$-function interaction of the same strength, the

S

matrix is explicitly given and shown to be symmetrical and unitary. The incoming and outgoing states may consist of bound compounds as well as single particles. The momenta of the particles and compounds are not changed in the scattering, but particles are exchanged, such as $ABC+DE\ensuremath{\rightarrow}ADC+BE$. Only distinguishable particles are considered.

The exact solution is presented of the scattering problem of three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.