We found that certain three-body excited states of (55Cs1H2)-Cs-133-H-3 in the Pd-12 cuboctahedron molecule could interfere with the three-nucleon resonance states of Cs+t+t or La-139(57)* which transfer from the 7/2(+) state of CsH2 to 7/2(+) state of La* by the E2 transition, and down to the La ground state, although the molecular ground state and the first excited state of (55Cs1H2)-Cs-133-H-3 are very stable. The three-body calculation is performed by using six potentials. We also found that the critical fusion value of t x rho x T (t:duration time, rho: plasma density, T: temperature or energy) for the ultra low energy molecular synthesis is almost the same as that for the thermal nuclear fusion, or higher. It is claimed that the radioactive nuclear waste products of the nuclear fission, could form via nuclear fusion a stable nucleus due to our ultra-low energy molecule-nucleus transition.

A new quantization is proposed for an energy-dependent particle exchange potential. The quantum number corresponds not only to the branch point of the potential cut, but also to the index of a long-range hadron potential. By using the long-range potential, an ultra-low energy nuclear (ULEN) reaction is investigated. It was found that a three-body Cs + H + H plasma in a Pd _12 cage could be excited to 1.36× 10^-4 eV ∼ 300 eV by an electric current where the long-range potential gives one or two order larger reaction probability than the others. A ULEN reaction was compared with the thermo-nuclear fusion by using well-known critical fusion constant defined by C_high/low = (a duration time) × (a plasma density) × (a temperature). We found that C_low is almost the same as C_high or more. We concluded that the ULEN synthesis could occur with a high-pressure hydrogen circumstance and a high-density CsH _2 Pd _12 cluster.

The three-body nuclear and molecular resonances for ^135_ 55 Cs+ ^2_1 H+ ^2_1 H, and ^133_ 55 Cs+ ^3_1 H+ ^3_1 H systems are calculated in “cuboctahedron ^135_ 55 Cs ^2_1 H _2 ^A_46 Pd _12 and ^133_ 55 Cs ^3_1 H _2 ^A_46 Pd _12 clusters” in a very wide range from 0.01[fm] to several hundreds of nm in “one stretch” with more than “100 significant figures”, where the mass number A of Pd could be 102, 104, 105, 106, 108, 110 but neglected hereafter, because Pd isn’t concerned directly with the nuclear reaction. We obtained several new “three-ion resonance states” between the expected molecular CsH _2 ground state and the first excited state in cuboctahedron CsH _2 Pd _12 cluster, where H represents either a ^1_1 H, a ^2_1 H, or a ^3_1 H, respectively. The molecular “ground and the first excited states” in the cluster are derived by the Kohn-Sham equation or the ADF package which could mainly describe many electrons rather than cores of ions. We found that the E2 transition times from some CsH _2 (7/2^+) resonance states (or the IOS states) to the nuclear ^139_ 57 La (7/2^+) ground state are about τ =10^-1∼ 10^-6 sec for five traditional potentials, and τ =10^-2∼ 10^-8 sec for six potentials with our long range three-body force (3BLF) where the “molecular resonances” can strongly interfere with the “nuclear resonances”. The thermal nuclear “critical reaction value” (or fusion constant) and/or ultra low energy corresponding value: C_high/low =(duration time) × (density) × (energy or temperature) are compared. It was found that C_low is almost the same order as C_high or more. Finally, an ignition method for the synthesis will be discussed.

The three-body energy levels of the 55135 CsD2 quasi-molecule in a CsD2Pd12 cluster are calculated using the Cs-D, D-D, Cs-Pd and D-Pd repulsive Coulomb interactions with electronic effects, the central Woods-Saxon potential for the two-body nuclear interactions, and a nuclear three-body potential term. The calculation was performed for distances ranging from fermis to nanometers, in one pass, using 100-figure numerical precision, and for energies between the 57139 La ground state (-25.92 MeV) to the top of the Pd-barrier. The ion bases calculated results are adjusted to the electron bases chemical ones at the ground state and at the first excited state of CsD2Pd12. The possibility of 55135 Cs(2D, γ) 57139 La reaction is investigated for the E2 transition between J π = 7/2+ of 55135 CsD2 and 7/2+ of the 57139 La ground state.

The general particle transfer (GPT) potential generates not only the Yukawa-type potential but also the 1/ r n - type potential in the hadron system, where the mass dependence of the transferred (exchanged) particle is clarified. The GPT potential from the atom-molecule system to the quark-gluon system was studied, where pico-meter physics could be highlighted. It is demonstrated that the long-range three-body Efimov potential is connected with the short-range three-body force potential by the GPT potential. Some applications for historical few-body problems in physics are summarized.

A long range potential the so-called general particle transfer (GPT) potential was proposed for three-body systems almost one decade ago. In this report, we will illustrate another GPT feature from the dispersion theoretical point of view. The index number of the long range potential is related to the exchange particle mass which represents the threshold of multi-pion creation in the potential cut. The index number is proportional to the exchange particle mass. Therefore, the potential with a large exchange particle mass is negligible compared to that with a small mass. Since the quasi-two-body form factors are also given by the GPT potential, then the GPT potential is represented not by a linear form but by an entangled non-linear long range potential. This results in a long range three-body Efimov-type potential.

We investigate a possibility of nuclear reaction near the three-body break-up threshold (3BT) by a viewpoint of the three-body Hamiltonian. We study a virtual molecule: CsD _2 which is well developed in a cuboctahedron CsD _2 Pd _12 -cluster, where CsD _2 is surrounded by the Pd _12 -cage. We found that the energy levels, which are obtained without the Coulomb repulsive potential, could be boosted up by the repulsive Coulomb correction, however the Coulomb excited resonant-level does not decay by the barrier of Pd _12 -cage. The boost up effect resolves a difficulty of the penetration problem. If the molecular level E_mol and the boosted up nuclear levels E_nucl(

A new potential, the generalgeneralparticleparticletransfertransferpotentialpotential: so called “GPT” potential was represented in the previous paper which indicates a Yukawa-type potential for shorter range, but a 1/r^n1/rn-type potential for longer range where n=2n=2 includes the Efimov-like potential in the hadron system. In order to confirm the existence of a GPT potential, we investigate the possibility of Cs+2d\rightarrow→La reaction on the three-ion quasi-molecule CsD_22 which is covered by twelve Pd or a CsD_22Pd_{12}12-cluster, where the three-body bound states and wave functions for D-Cs-D molecular and d-Cs-d nuclear systems are calculated. We obtain an approximate E2-transition from the molecular states to the nuclear states. The transition ratio between the short range nuclear potential with the 1/r^21/r2-type long range potential and without long range potential is W_{i\rightarrow f}^{E2';L}/W_{i\rightarrow f}^{E2';S}\approx 10^8Wi→fE2′;L/Wi→fE2′;S≈108. If the reaction Cs+2D\rightarrow→La is experimentally observed, then the existence of the GPT potential could be confirmed.

The existence of a kinematic long range component in the one particle transfer three-body potential or so-called “general particle transfer (GPT) potential” was proposed several years ago. In this investigation the mass dependence of the exchanged particle and the index number of the long range property are clarified. On the basis of the GPT potential, a new long range three-body force is proposed in the hadron system, which will be compared with the Efimov potential.

We investigate the possibility of lanthanum (La)-nucleus creation via the reaction Cs(2d, $$\gamma $$ )La on the three-ion quasi-molecule $$\hbox {CsD}_2$$ in the $$\hbox {CsD}_2\hbox {Pd}_{{12}}$$ -cluster. In order to calculate d–Cs–d (or D–Cs–D) three-body bound states and wave functions, we adopt a very accurate three-body variational method with more than 80 figures. The $$\hbox {CsD}_2$$ and La binding energies and wave functions are obtained with an S-wave trial functions. The wave function overlap (WFO) value between the La nuclear excited state and the several $$\hbox {CsD}_2$$ quasi molecular modified states is calculated. It should be stressed that the WFO value is of critical importance for the existence of the electro-magnetic transition in the Cs(2d, $$\gamma $$ )La reaction. We found that the WFO value is very sensitive to the nuclear potential tail with a two- or three-body $$1/r^2$$ -type long range hadron potential, and gives a promising result for the reaction. The result is also followed by the three-body Faddeev calculation.

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: \(A+(BC)\rightarrow A+B+C\), and \((ABC)\rightarrow A+(BC)\), 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 \(1/r^n\)-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: \(E_n=\epsilon +\zeta _n\) with the two-body binding energy \(\epsilon \), and the separation energy \(\zeta _n\) for \((ABC)\rightarrow A+(BC)\) satisfies \(E_n/E_{n+1}=\zeta _n/\zeta _{n+1}\)=const for \(\epsilon =0\) or the two-body scattering length: \(a\rightarrow \infty \) (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 \(E_n/E_{n+1}=(\zeta _n+\epsilon )/(\zeta _{n+1}+\epsilon )\rightarrow 1\) (const) for \(n\rightarrow \infty \) (with \(\zeta _n\rightarrow 0\)) 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.

We confirm the reliability of the well-known Coulomb renormalization method (CRM). It is found that the CRM is only available for a very-long-range screened Coulomb potential (SCP). However, such an SCP calculation in momentum space is considerably difficult because of the cancelation of significant digits. In contrast to the CRM, we propose a new method by using an on-shell equivalent SCP and the rest term. The two-potential theory with r-space is introduced, which defines fully the off-shell Coulomb amplitude.

A Coulomb equivalent screened Coulomb potential is proposed for solving the Schrödinger equation and/or the Calogero first order differential equation, where some critical range bands are obtained. Phase shifts for “any” two-charged particle system (from electron–electron to heavy ion–heavy ion) are reproduced by using the universal critical range bands and the appropriate Sommerfeld parameter over a very wide energy region. A Coulomb-like off-shell amplitude is introduced using two-potential theory without employing the usual Coulomb renormalization method.

We propose an off-shell Coulomb-like amplitude with an on-shell phase shift that is accurate to within 9-10 digits. The full Coulomb amplitude is separated into the leading and the auxiliary amplitudes using “the two-potential theory”. The leading amplitude reproduces most of the on- and off-shell parts, while the auxiliary amplitude contributes mainly to the off-shell part and minimally to the on-shell part. We then review the three-body calculation method developed during the last four decades. A reminder to a threshold behavior investigation method is pointed out based on the three-body Faddeev equation. We discuss the Efimov physics and some extensions, which are recovered from our predictions. The method may suggest a promising technique to resolve the existing discrepancies between current experimental and theoretical values.

n α – n – n three-cluster model of the ^6 He nucleus is studied by solving the Faddeev equations, where the cluster potential between α and n takes into account the Pauli exclusion correction, using the Fish-Bone Optical Model (Schmid in Z Phys A 297:105, 1980 ). The resulting binding energy of the ground state ( 0^+ ) is 0.831 MeV and the resonance energy of the first excited state ( 2^+ ), 0.60–i0.012 MeV, is extracted from the three-cluster break-up threshold. These theoretical values are in reasonable agreement with the experimental data: 0.973 MeV and 0.824–i0.056 MeV, respectively. In order to investigate the structure of these states, we calculate the angle density matrix for the ∠ n_1 α n_2 angle in the triangle formed by the three clusters. The angle density matrix of the ground state has two peaks and the configuration of 0^+ wave function corresponding to the peaks constitutes a mixture of an acute-angled triangle structure and an obtuse-angled one. This finding is consistent with the former result from a variational approach (Hagino and Sagawa in Phys Rev C 72:044321, 2005 ). On the other hand, in the case of 2^+ state only a single peak is obtained.

An \({\alpha}\)–n–n three-cluster model of the \({^6}\)He nucleus is studied by solving the Faddeev equations, where the cluster potential between \({\alpha}\) and n takes into account the Pauli exclusion correction, using the Fish-Bone Optical Model (Schmid in Z Phys A 297:105, 1980). The resulting binding energy of the ground state (\({0^+}\)) is 0.831 MeV and the resonance energy of the first excited state (\({2^+}\)), 0.60–i0.012 MeV, is extracted from the three-cluster break-up threshold. These theoretical values are in reasonable agreement with the experimental data: 0.973 MeV and 0.824–i0.056 MeV, respectively. In order to investigate the structure of these states, we calculate the angle density matrix for the \({\angle n_1 \alpha n_2}\) angle in the triangle formed by the three clusters. The angle density matrix of the ground state has two peaks and the configuration of \({0^+}\) wave function corresponding to the peaks constitutes a mixture of an acute-angled triangle structure and an obtuse-angled one. This finding is consistent with the former result from a variational approach (Hagino and Sagawa in Phys Rev C 72:044321, 2005). On the other hand, in the case of \({2^+}\) state only a single peak is obtained.

We confirmed that the Coulomb phase shifts can not be given with a screened Coulomb potential by a monotonic increase of the range R. However, several screening “discrete range-bands” reproduce the Coulomb phase shift in a wide energy region. If the range-band of a proton-proton system for example is obtained, then the range-bands are generalized by three universal arguments: the universal range R(k)(≡ kR), the Sommerfeld parameter η(k)(≡ Z1Z2e2ν/k), and the universal asymptotic phase F(k)(≡ 2krc). Finally, the phase shifts of any other systems from e−−e− to heavy ions such as 208Pb−208Pb can be reproduced automatically by an individual range R = R(k)/k on the universal range-bands R(k).

The two-body threshold behavior at NN′ and πD are investigated by using the multi-channel Lippmann-Schwinger equations with an energy dependent two-body quasi potential, which are analytically continued from the three-body Faddeev equations at the three-body break up threshold. Our calculated NN′ and πD scattering lengths show better agreement with the experimental data for NN and πD systems than those from the original NNπ three-body Faddeev equations.

An -n-n three-cluster model of the He nucleus is studied by solving the Faddeev equations, where the cluster potential between and n takes into account the Pauli exclusion correction, using the Fish-Bone Optical Model (Schmid in Z Phys A 297:105, 1980). The resulting binding energy of the ground state () is 0.831 MeV and the resonance energy of the first excited state (), 0.60-i0.012 MeV, is extracted from the three-cluster break-up threshold. These theoretical values are in reasonable agreement with the experimental data: 0.973 MeV and 0.824-i0.056 MeV, respectively. In order to investigate the structure of these states, we calculate the angle density matrix for the angle in the triangle formed by the three clusters. The angle density matrix of the ground state has two peaks and the configuration of wave function corresponding to the peaks constitutes a mixture of an acute-angled triangle structure and an obtuse-angled one. This finding is consistent with the former result from a variational approach (Hagino and Sagawa in Phys Rev C 72:044321, 2005). On the other hand, in the case of state only a single peak is obtained.