Operation of the Faddeev Kernel in Configuration Space

We present a practical method to solve Faddeev three-body equations at energies above the three-body breakup threshold as integral equations in coordinate space. This is an extension of a previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.     

Three-Potential Formalism for the Atomic Three-Body Problem

Based on a three-potential formalism we propose mathematically well-behaved Faddeev-type integral equations for the atomic three-body problem and describe their solutions in Coulomb-Sturmian space representation. Although the system contains only long-range Coulomb interactions these equations allow us to reach a solution by approximating only some auxiliary short-range potentials. We outline the method for bound states and demonstrate its power in benchmark calculations. We can report a fast convergence in angular-momentum channels. Received September 30, 1997; accepted in final form January 23, 1998     

On the motion of classical three-body system with consideration of quantum fluctuations

We obtained the systemof stochastic differential equations which describes the classicalmotion of the three-body system under influence of quantum fluctuations. Using SDEs, for the joint probability distribution of the total momentum of bodies system were obtained the partial differential equation of the second order. It is shown, that the equation for the probability distribution is solved jointly by classical equations, which in turn are responsible for the topological peculiarities of tubes of quantum currents, transitions between asymptotic channels and, respectively for arising of quantum chaos.     

Dynamically generated N~* resonances from the interaction of two mesons and a baryon

We have studied the ππN system and coupled channels by using of the Faddeev equations and two N~* and one △ states, all of them with J~P = 1/2~+, have been found in the formalism as dynamically generated states. In addition, signatures for a new N* resonance with J~P = 1/2~+ are found around an energy of 1920 MeV in the three-body center of mass system.     

Accumulation of three-body resonances above two-body thresholds

We calculate resonances in the three-body system with attractive Coulomb potential by solving homogeneous Faddeev-Merkuriev integral equations for complex energies. The equations are solved using the Coulomb-Sturmian separable expansion method. This approach allows us to study the exact behavior of the three-body Coulomb systems near the threshold. A negatively charged positronium ion is used as a test case. In addition to locating all previously known S-wave resonances of the positronium ion, we also find a large number of new resonant states that accumulate just slightly above the two-body thresholds. The pattern of accumulation of resonant states above the two-body thresholds suggests that probably they are infinite in number. We conjecture that this may be a general property of the three-body system with an attractive Coulomb potential.     

The similarity renormalization group for three-body interactions in one dimension

We report on recent progress of the implementation of the similarity renormalization group (SRG) for three-body interactions in a one-dimensional, bosonic model system using the plane-wave basis. We discuss our implementation of the flow equations and show results that confirm that results in the three-body sector remain unchanged by the transformation of the Hamiltonian. We also show how the SRG transformation decouples low- from high-momentum nodes in the three-body sector and therefore simplifies the numerical calculation of observables.     

Three-body model for neutron-halo nuclei

The neutron-halo nuclei, ¹¹Li, ¹⁴Be, and ¹⁷B, are studied in the three-body model. The Yukawa interaction is used to describe the interaction of the two-body subsystem. For given parameters of the two-body interaction, the properties of these neutron-halo nuclei are calculated with the Faddeev equations and the results are compared with those in the variational method. It is shown that the method of the Faddeev equations is more accurate. Then the dependencies of the two- and three-body energies on the parameters are studied. We find numerically that two- and three-body correlations differ greatly from each other with the variation of the intrinsic force range.     

Three-body model for neutron-halo nuclei

The neutron-halo nuclei,11Li,14Be,and 17B,are studied in the three-body model.The Yukawa interaction is used to describe the interaction of the two-body subsystem.For given parameters of the two-body interaction,the properties of these neutron-halo nuclei are calculated with the Faddeev equations and the results are compared with those in the variational method.It is shown that the method of the Faddeev equations is more accurate.Then the dependencies of the two-and three-body energies on the parameters are studied. We find numerically that two-and three-body correlations differ greatly from each other with the variation of the intrinsic force range.     

Effective Field Theory and the Gamow Shell Model

We combine halo/cluster effective field theory (H/CEFT) and the Gamow shell model (GSM) to describe the 0⁺ ground state of ⁶He as a three-body halo system. We use two-body interactions for the neutron-alpha particle and two-neutron pairs obtained from H/CEFT at leading order, with parameters determined from scattering in the p_3/2 and s₀ channels, respectively. The three-body dynamics of the system is solved using the GSM formalism, where the continuum states are incorporated in the shell model valence space. We find that in the absence of three-body forces the system collapses, since the binding energy of the ground state diverges as cutoffs are increased. We show that addition at leading order of a three-body force with a single parameter is sufficient for proper renormalization and to fix the binding energy to its experimental value.     

Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.     

Integral Equations for Three-Body Coulomb Resonances

 We propose a novel method for calculating resonances in three-body systems with repulsive Coulomb interactions. The method is based on the solution of a set of Faddeev and Lippmann-Schwinger integral equations. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order to get the resonances. As a numerical illustration a model for the three-α system is solved. Received October 1, 1999; revised February 25, 2000; accepted for publication June 30, 2000     

Low-Energy Proton-Deuteron Scattering in Configuration Space

 A new method for solving the configuration-space Faddeev equations for elastic p-d scattering below the deuteron-breakup threshold is described. Numerical solutions that demonstrate the convergence and accuracy of the method are given. The number of channels and the value of the matching radius required to obtain an accurate solution are also investigated. These calculations demonstrate that this method can efficiently solve the large matrix equations required for the three-body scattering problem. Received April 23, 2001; accepted for publication June 7, 2001     

S = −1 resonances in two meson-one baryon systems

We study the two meson-one baryon systems by solving Faddeev equations, using chiral dynamics. The calculations, carried out for the $\pi \overline{K} N$ system and its coupled channels for the case of strangeness = ?1, in the S-wave, lead to a dynamical generation of many strangeness = ?1 resonances in the 1500–1800?MeV region. While building the formalism, we found important cancellations between different sources of three-body forces.     

Stability and Chaos of Two Coupled Bose-Einstein Condensates with Three-Body Interaction

We study the dynamics of two Bose-Einstein condensates (BECs) tunnel-coupled by a double-well potential.A real three-body interaction term is considered and a two-mode approximation is used to derive two coupled equations,which describe the relative population and relative phase. By solving the equations and analyzing the stability of the system, we find the stable stationary solutions for a constant atomic scattering length. When a periodically time-varying scattering length is applied, Melnikov analysis and numerical calculation demonstrate the existence of chaotic behavior and the dependence of chaos on the three-body interaction parameters.     

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.     

Three-body inverse scattering problem

Two methods are suggested to reconstruct three-body potentials from three-body scattering data. This was achieved by using the reduction of the corresponding Schrödinger equation to a system of ordinary differential equations (not integro-differential equations as usual in the direct problem). Exactly solvable three-body models are presented. A new simple method for solving the multi-dimensional inverse problem in a finite-difference approximation is considered in the Appendix.     

Three-Body Problem with Two Charged Particles (I) The Neutron Induced Nuclear Reaction

We present the details of an exact method for the treatment of Coulomb effects in neutron induced three-body n uclear reactions. Based on the three-body in tegral equations, the formalist allows the practical calculation of elastic, inelastic, and breakup processes without an approxima tion of C.M. Coulomb interaction.     

Universality in the Four-Body System in an EFT Framework

We extend previous calculations by Platter et al. to include higher partial waves using the same effective quantum mechanics approach for a non-relativistic four-boson system. The Yakubovsky equations are reduced to a simple eigenvalue problem, which is then solved numerically. Instead of introducing a three-body force, we fix the cutoff so that the three-body system is reproduced correctly. The thus obtained results show good agreement with Platter’s. Preliminary results for higher partial waves are shown as well.