A Three-body model for the study of resonance phenomena. III

The potential and quasi-compound resonances appearing in a model three-body system consisting of two identical light particles and an infinitely heavy one is investigated by solving the Faddeev-equations for negative total energy.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

A model for short range correlation in the three-body system

A method by means of which the two-body short-range correlations in the nucleon-nucleon interaction are taken accurately into account in three-body bound-state calculations is described. The method is independent of the nucleon-nucleon interaction. We employ our method to calculate meson-exchange corrections to the magnetic moments of ³H and ³He and the Coulomb energy of ³He.     

A three-body optical model for two clusters and a meson

Conclusion In a very simplified -²⁰Ne scattering calculation with the fish bone optical model, we have found a big difference between results obtained with two different off-shell versions of the model, when the three-body Pauli potential is neglected. It seems to be safe to say that the three-body force may not be neglected in the M-model. Further calculations will be needed to see whether it may be neglected in the ¯M-model.Presented at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.     

Efimov effect in nuclear three-body resonance decays

We investigate the effects of the nearly fulfilled Efimov conditions on the properties of three-body resonances. Using the hyper-spheric adiabatic expansion method we compute energy distributions of fragments in a three-body decay of a nuclear resonance. As a realistic example we investigate the 1- state in the halo nucleus 11Li within a three-body model 9Li + n +n model. Characteristic features appear as sharp peaks in the energy distributions. Their origin, as in the Efimov effect, is in the large two-body s-wave scattering lengths between the pairs of fragments.     

A variation-iteration method for the three-body problem

We construct a practicable formulation of the variation-iteration method for the calculation of properties of a tri-nucleon system interacting via a modern potential. Our general approach is to take some integral form of the Schrödinger equation and reduce it to a set of simpler integral equations which may be solved by iteration. We concentrate particularly on the simplicity of the kernels which appear in this last set. The best method appears to be a two-step one, essentially equivalent to the Green function. In constructing the formalism, we are led to define a set of functions $\text{?}{}_{\mathrm{MN}}{}^{\mathrm{L}}\text{(k}{}_{12}\text{, k}{}_{23}\text{, k}{}_{31}\text{, r}{}_{12}\text{, r}{}_{23}\text{, r}{}_{31}\text{)}$ which play the role of iteration kernels. We give various properties of $\text{?}{}_{\mathrm{MN}}{}^{\mathrm{L}}$ and indicate very briefly how $\text{?}{}^{\mathrm{L}}{}_{\mathrm{MN}}$ may be reduced to a function of only two variables.     

A model calculation in a new approach to the three-body problem

A method to calculate the binding energy of a system of three identical particles is proposed. The kernel of a previously derived integral equation in two variables is replaced by an approximation of finite rank. It turns out that a one term approximation is already sufficient to determine the binding energy very well.     

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.     

Delta function model for the helium trimer: origin of three-body forces

The Dirac bubble potential, previously used to model the helium dimer, is applied to the trimer. It is shown that the quantum mechanical kinetic energy operator for a three-body system contains terms over and above the analogues of classical pairwise contributions. The additional terms, the ‘Borromean couplings’, are responsible for a dramatic increase in the binding energy of the trimer. For example, ⁴He₃ is bounded by two orders of magnitude more strongly than ⁴He₂, 279 mK versus 1.31 mK, respectively, according to our calculations. Moreover, the trimer is considerably more compact than the dimer, with (r₁₂) decreasing to 9.01 Å from 51.9 Å.     

Interaction and resonance phenomena of multi-soliton

As is well known, Korteweg-de Vries equation is a typical one which has planar solitary wave. By considering higher order transverse disturbance to planar solitary waves, we study a Kadomtsev-Petviashvili (KP) equation and find some interesting results. In this letter we investigate the three soliton interaction and their resonance phenomena of KP equation, and theoretically find that the maximum amplitude is 9 times of the initial interacting soliton for three same amplitude solitions. Three arbitrary amplitude solition interaction of KP equation is also studied by numerical simulation, which can also results in resonance phenomena.     

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.     

A simple quark model for resonance electroproduction

A quark model for resonance electroproduction is developed which is based on the successful photoproduction model of Kubata and Ohta. A detailed comparison is made between the predictions of the model and the results of recent phenomenological analyses of electroproduction data.     

Self-propelled particle model for cell-sorting phenomena

A self-propelled particle model is introduced to study cell sorting occurring in some living organisms. This allows us to evaluate the influence of intrinsic cell motility separately from differential adhesion with fluctuations, a mechanism previously shown to be sufficient to explain a variety of cell rearrangement processes. We find that the tendency of cells to actively follow their neighbors greatly reduces segregation time scales. A finite-size analysis of the sorting process reveals clear algebraic growth laws as in physical phase-ordering processes, albeit with unusual scaling exponents.     

Relations between differential cross sections for three-body reactions from the quark model

In this paper relations between the differential cross sections for three-body processes following from the non-relativistic quark model are classified and compared with experimental data. In general, within the experimental errors, they show satisfactory agreement, except relations for strangeness-exchange processes, namely so called SU(6) relations and relations following from the assumption of identity of quarks constituting mesons and baryons. The relation of these predictions to those of other models, and this investigation to a previous analysis of quasi-two-body reactions in the quark model is discussed shortly.     

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.     

Stochastic resonance phenomena in spin chains

We discuss stochastic resonance-like effects in the context of coupled quantum spin systems. We focus here on an information-theoretic approach and analyze the steady state quantum correlations (entanglement) as well as the global correlations in the system when subject to different forms of local decoherence. In the presence of decay, it has been shown that the system displays quantum correlations only when the noise strength is above a certain threshold. We extend this result to the case of a Heisenberg XYZ exchange interaction and revise and clarify the mechanisms underlying this behaviour. In the presence of pure dephasing, we show that the system always remains separable in the steady state. When both types of noise are present, we show that the system can still exhibit entanglement for long times, provided that the pure dephasing rate is not too large.     

One-dimensional Bose-Hubbard model with pure three-body interactions

The extended Bose-Hubbard model with pure three-body local interactions is studied using the Density Matrix Renormalization Group approach. The shapes of the first two insulating lobes are discussed, and the values of the critical tunneling for which the system undergoes the quantum phase transition from insulating to superfluid phase are predicted. It is shown that stability of insulating phases, in contrast to the standard Bose-Hubbard model, is enhanced for larger fillings. It is also shown that, on the tip of the boundary of the insulating phase, the model under consideration belongs to the Berenzinskii-Kosterlitz-Thouless universality class.