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Our approach aims at a general formalism for the quantum description of the three-body Coulomb systems. We seek the exact solutions of 6D Schrödinger equation. For this, we propose a new algorithm for the case of nonzero total angular momentum, taking into account the overall rotation of the system, which is affected indirectly by the Coriolis coupling. We construct a special set of hyperspherical harmonics, which provide much more flexibility in choosing the best basis for the needs of this particular physical problem. The robustness, efficiency, and accuracy of the adopted algorithm are studied in detail. We apply this method to the computation of the nonrelativistic energy levels of the exotic helium .

We exploit the possibility of new configurations in three-body halo nuclei, Samba type (the neutron-core form a bound system) as a doorway to Borromean systems. The nuclei ¹²Be, ¹⁵B, ²³N and ²⁷F are of such nature, in particular ²³N with a half-life of 37.7 s and a halo radius of 6.07 fm is an excellent example of Samba-halo configuration. The fusion below the barrier of the Samba halo nuclei with heavy targets could reveal the so far elusive enhancement and a dominance of one-neutron over two-neutron transfers, in contrast to what was found recently for the Borromean halo nucleus ⁶He+²³⁸U.

Study of Coulombian three-body system is a basic phenomenon in muon catalyzed fusion (μCF). In this investigation, separation of variables in the base of adiabatic expansion, have been applied to the mesic three-body molecule, ³Heμd using hyper-spherical elliptic coordinate system. The corresponding eigenvalue problem has been solved and the adiabatic potential and the binding energy of this system are calculated. The obtained results agreed with the expected values of various theoretical methods.

The radial wave functions of the Bear–Hodgson potential have been used to study the ground state features such as the proton, neutron and matter densities and the associated rms radii of two neutrons halo ⁶He, ${}^{11}$Li, ${}^{14}$Be and ${}^{17}$B nuclei. These halo nuclei are treated as a three-body system composed of core and outer two-neutron $(\mathrm{\text{Core}}+n+n)$. The radial wave functions of the Bear–Hodgson potential are used to describe the core and halo density distributions. The interaction of core-neutron takes the Bear–Hodgson potential form. The outer two neutrons of ⁶He and ${}^{11}$Li interact by the realistic interaction REWIL whereas those of ${}^{14}$Be and ${}^{17}$B interact by the realistic interaction of HASP. The obtained results show that this model succeeds in reproducing the neutron halo in these nuclei. From the calculated densities, it is found that ⁶He, ${}^{11}$Li, ${}^{14}$Be and ${}^{17}$B have a long tail in neutron and matter densities which is consistent with the experimental data. Elastic charge form factors for these halo nuclei are analyzed via the plane wave Born approximation.

The group theoretical description of the three-particle problem provides successful techniques for the solution of different questions. We present here a review of this approach.

We consider the classical three-body system with $d$ degrees of freedom $(d>1)$ at zero total angular momentum. The study is restricted to potentials $V$ that depend solely on relative (mutual) distances $r_{ij}=|r_i-r_j|$ between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on $d$, confirming results by Murnaghan (1936) at $d=2$ and van Kampen–Wintner (1937) at $d=3$, where it corresponds to a 3D solid body. Realizing ${\mathbb{Z}}_2$-symmetry $(r_{ij}\to -r_{ij})$, we introduce new variables ${\rho }_{ij}=r_{ij}^2$, which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the $\rho$-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is ${{𝒮}}_3$-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of ${\mathbb{Z}}_2^{\otimes 3}\oplus {{𝒮}}_3$ to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple.

We study three examples in some detail: (I) three-body Newton gravity in $d=3$, (II) three-body choreography in $d=2$ on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

What is the origin of macroscopic randomness (uncertainty)? This is one of the most fundamental open questions for human beings. In this paper, 10 000 samples of reliable (convergent), multiple-scale (from 10^-60 to 10²) numerical simulations of a chaotic three-body system indicate that, without any external disturbance, the microscopic inherent uncertainty (in the level of 10^-60) due to physical fluctuation of initial positions of the three-body system enlarges exponentially into macroscopic randomness (at the level O(1)) until t = T*, the so-called physical limit time of prediction, but propagates algebraically thereafter when accurate prediction of orbit is impossible. Note that these 10 000 samples use micro-level, inherent physical fluctuations of initial position, which have nothing to do with human beings. Especially, the differences of these 10 000 fluctuations are mathematically so small (in the level of 10^-60) that they are physically the same since a distance shorter than a Planck length does not make physical sense according to the string theory. This indicates that the macroscopic randomness of the chaotic three-body system is self-excited, say, without any external force or disturbances, from the inherent micro-level uncertainty. It provides us the new concept "self-excited macroscopic randomness (uncertainty)". The macroscopic randomness is found to be dependent upon microscopic uncertainty, from the statistical viewpoint. In addition, it is found that, without any external disturbance, the chaotic three-body system might randomly disrupt with symmetry-breaking at t = 1000 in about 25% probability, which provides us new concepts "self-excited random disruption", "self-excited random escape" and "self-excited symmetry breaking" of the chaotic three-body system. Hence, it suggests that a chaotic three-body system might randomly evolve by itself, without any external forces or disturbance. Thus, the world is essentially uncertain, since such kind of self-excited macroscopic randomness (uncertainty) is inherent and unavailable. This work also implies that an universe could randomly evolve by itself into complicated structures, without any external forces. To emphasize this point, the so-called "molecule-effect" (or "nonbutterfly effect") of chaos is suggested in this paper. All of these reliable computations could deepen our understandings of chaos from physical viewpoints, and reveal a kind of origin of macroscopic randomness/uncertainty.

Lattice QCD simulation for hadron spectroscopy has reached three-body threshold. However, it is difficult to go further without an appropriate method to analyze the finite volume effect in three-body system. We derive the three-body quantization condition in a finite volume using an effective field theory in the particle-dimer picture. It is a powerful and transparent method to interpret three-body physical observables from lattice simulations. We review the whole method including formalism, quantization condition and the application for both 3-body bound states and scattering states.

We investigate the three-body systems of and , by taking the fixed center approximation to Faddeev equations. We find a clear and stable resonance structure around 1490 MeV in the scattering amplitude for the system, which is not sensitive to the renormalization parameters. This resonance is associated to the η(1475). We get only an enhancement effect of the threshold in the amplitude.