Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.     

On the three-body Coulomb scattering problem

We describe the smoothness properties and the asymptotic form of the Green's function (in configuration space) for three charged particles. We also discuss the integral equations and the boundary value problems for the Coulomb wavefunctions and we show that they form a complete set. Finally, we study the singularities of the Coulomb scattering operator, and we investigate the connection between the Dollard wave operators and the Coulomb wavefunctions.     

Variational calculations on the three-body scattering problem

An extended resonating-group method is used to calculate the elastic scattering amplitudes (up to L = 2 for a system of three identical bosons interacting through local Yukawa potentials. The results are compared to approximate solutions of the Faddeev equations.     

A three-body scattering problem in the molecular mass limit

A three-body problem is studied that involves the scattering of a heavy particle from a bound state of a heavy and a light particle. For fairly large mass ratios we find many partial waves involved in the scattering with both elastic and inelastic resonances present. It is also found that virtually all of the breakup inelasticity is in the odd partial waves, if the heavy particles are identical bosons. The same problem is solved in the Born-Oppenheimer approximation and it is found that the effective potential between the heavy particles develops an imaginary part in the odd partial waves and thus gives a quantitative account of the three-body results including the inelasticity. The linear combination of nuclear orbitais method is applied to the same problem and is found to be inadequate.     

A non-singular equation for the three-body scattering problem

This paper derives a non-singular integral equation for the three-body problem. Starting from the three-body equations obtained by Karlsson and Zeiger we introduce a set of algebraic transformations that remove all the Green function pole singularities. For scattering energies on the real axis we find a singularity-free momentum-space integral equation. This equation requires only a finite range of momentum values for its solution. In the case of well-behaved two-body interactions, such as the superposition of Yukawa interactions, we prove that the kernels of this equation have a finite Hilbert-Schmidt norm. This same norm provides a general criteria for establishing when the impulse approximation is accurate.     

Potential splitting approach to the three-body coulomb scattering problem

The potential splitting approach is extended to a three-body Coulomb scattering problem. The distorted incident wave is constructed and the driven Schrödinger equation is derived. The full angular momentum representation is used to reduce the dimensionality of the problem. The phase shifts for e⁺?H and e⁺?He⁺ collisions are calculated to illustrate the efficiency of the presented method.     

A novel method for solving the three-body scattering problem

A numerical scheme for solving a three-body scattering problem within the framework of the configuration space Faddeev equations in three-dimension, i.e., without resort to explicit partial wave expansion, is presented. The method is applied to calculate the low-energy n-d observables.     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.     

New effective adiabatic approach to the muonic three-body scattering problem

The effective adiabatic approach to analysis of the three-particle interaction is presented. It gives a possibility to represent even in a simple two-level approximation all qualitative peculiarities of mesic atomic resonance reactions and to obtain a good quantitative agreement with different cumbersome calculations.     

Pion-deutron scattering in the Δ(1236) energy region as a three-body problem

We study pion-deuteron scattering in the first π-N resonance energy region, in a three-body model based on Faddeev's equations. We discuss the effects of multiple scattering, nuclear binding, and virtual excitation of the target, on the energy behavior of the cross section.     

On the three-body long-range scattering problems

In this Letter, we give results on precise microlocalized time-decay estimates in three-body long-range scattering problems. We prove the asymptotic completeness of wave operators in three-body long-range scattering for a class of long-range interactions of the form V ₁(x)+V ₂(x), where V ₁ is nonnegative and decays like O(|x|^–0), for some ₀ > 1/2 and V ₂ decays like O(|x|^-y) for some > 2(1–₀)/₀.     

Multiple scattering with three-body forces

The multiple-scattering formalism is generalized to account for simultaneous interactions of a projectile with two target particles. In the cluster expansion of multiple-scattering theory such three-body interactions combine with iterates of two-body interactions. Some comparisons of multiple scattering with the Faddeev theory are given.     

Sundman stability in the general three-body problem

A new method for studying the Hill-type stability in the general three-body problem using Sundman??s inequality is presented. Sundman??s surfaces in 3D space are constructed, which are counterparts of Hill??s surfaces. The conditional and unconditional Sundman stability criteria are established and used for determining the stability regions.     

Alternative treatment of the three-body problem

For solving the three-body problem with local potentials a model HamiltonianH ₀ containing an interaction between one particle and the centre-of-mass of the other two interacting particles is introduced. The total HamiltonianH is obtained byH=H _o +W whereW is a “residual interaction” in close analogy to the nuclear shell model. At a certain stage of the calculationsH ₀ has to be replaced by a new model Hamiltonian \(\tilde H_0 \) containing plane waves. The resolvent (and thereby theT-matrix) of the three-body problem is calculated by operator techniques. It is possible to draw some conclusions concerning three-body properties from these general expressions. Therefore this attempt may be considered as a supplementary treatment, in addition to the Faddeev-equations, of the three-body problem: it exhibits the discrete spectrum, the simple and the twofold continuum ofH arising from the corresponding states ofH ₀, and provides some approximation methods.     

The three-body problem in the path integral formalism

The propagator related to the Calogero potential is calculated in the phase space by way of Feynman formalism. The energy spectrum is determined along with the corresponding wave functions. In case some constraints are introduced, the problem may be reduced to the one corresponding to a particle constrained to move into a sector of opening angle α. It is shown as well that complicated potentials, may be transformed to allow the calculation of the energy spectrum via the Kleinert method.     

Model three-body problem in the molecular mass limit

The bound states of a three-body molecule composed of two identical heavy nuclei and a light “electron” interacting through short-range s-wave potentials are studied. The spectrum of three-body bound states grows as the mass ratio m between the heavy and light particles increases, and presents a remarkable vibration rotation structure that can be fitted with the usual empirical energy formulas of molecular spectroscopy. The results of the exact three-body calculation for the binding energy and bound-state wavefunction are compared with the predictions of the Born-Oppenheimer method for the same system. We find that for m > 30, the Born-Oppenheimer approximation yields very good results for both the binding energies and wavefunctions. For smaller m (1 <m < 30) the Born-Oppenheimer results are still surprisingly good and this is shown to be related to the range of the two-body interactions.     

Discrete Newtonian gravitation and the three-body problem

Newtonian gravitation is studied from a discrete point of view, in that the dynamical equation is an energy-conserving difference equation. Application is made to planetary-type, nondegenerate three-body problems and several computer examples of perturbed orbits are given.     

Choreographic solution to the general-relativistic three-body problem

We reexamine the three-body problem in the framework of general relativity. The Newtonian N-body problem admits choreographic solutions, where a solution is called choreographic if every massive particle moves periodically in a single closed orbit. One is a stable figure-eight orbit for a three-body system, which was found first by Moore (1993) and rediscovered with its existence proof by Chenciner and Montgomery (2000). In general relativity, however, the periastron shift prohibits a binary system from orbiting in a single closed curve. Therefore, it is unclear whether general-relativistic effects admit choreography such as the figure eight. We examine general-relativistic corrections to initial conditions so that an orbit for a three-body system can be choreographic and a figure eight. This illustration suggests that the general-relativistic N-body problem also may admit a certain class of choreographic solutions.