We consider a system of three one-dimensional particles with one of pair potentials V α ( x α ) decaying at infinity as
∣xα∣−ρ,0<ρ<1/2 . It is shown that such a system can possess channels of scattering not included in the usual list of channels called the asymptotic completeness. Similar channels arise in a two-body system with a long-range potential if a traditional condition on its derivatives is relaxed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
On the three-body kinematics, we investigate the threshold behavior which appears not only at the three-body break-up threshold (3BT), but also at the quasi two-body threshold (Q2T) for the reactions:
A+(BC)→A+B+C , and
(ABC)→A+(BC) , respectively. Recently, the author proposed a general particle
transfer (GPT) potential which appears, not only at the 3BT, but also at the Q2T between A and (BC). The new potential indicates a Yukawa-type potential for short range, but a
1/rn -type potential for long range. The long range part of the GPT potential for
n=1 indicates an attractive Coulomb-like or a gravitation-like potential. While,
n=2 indicates the Efimov-like potential between A and (BC). The three-body binding energy:
En=ϵ+ζn with the two-body binding energy
ϵ , and the separation energy
ζn for
(ABC)→A+(BC) satisfies
En/En+1=ζn/ζn+1 =const for
ϵ=0 or the two-body scattering length:
a→∞ (i.e. the two-body unitary limit). At the Q2T, the condensation of the three-body binding energy is given by the GPT-potential in the form of
En/En+1=(ζn+ϵ)/(ζn+1+ϵ)→1 (const) for
n→∞ (with
ζn→0 ) which implies the existence of Efimov-like states at the Q2T in the hadron systems, thereby the possibility of “ultra low energy nuclear transformation”, where the three-body binding energies degenerate at zero energy. Finally, the origin of such a long range potential will be clarified.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.