# The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems

## Abstract

**:**

## 1. Introduction

## 2. The Modified Faddeev Equation

_{α}(x

_{α}, y

_{α}) vanishes asymptotically within the three-body sector, where both x

_{α}, y

_{α}→ ∞ and approaches one in the two-body cluster region, where x

_{α}$\ll $ y

_{α}→ ∞.

_{0}, y

_{0}must be chosen very carefully, more details are presented in Section 3.

_{3}(x

_{3}) be repulsive, we can choose ζ

_{3}(x

_{3}, y

_{3}) ≡ 0. Such that the MFE is a set of two coupled differential equations. The total wavefunction of the scattering system is $\psi ={\sum}_{\alpha =1}^{2}{\psi}_{\alpha}$, α = 1, 2 are the two Faddeev channels.

## 3. Numerical Method

- $\nu =2.1$ for both Faddeev channels produced the most stable solutions.
- Both ${x}_{0}$, ${y}_{0}$ depend on the respective Faddeev channels. Since ${x}_{0}$ is proportional to the size of the two-body bound states and ${y}_{0}$ is related to the cutoff parameters, ${y}_{max}$.

^{+}+ H (n = 1)

e

^{+}+ H (n = 2, l = 0)

e

^{+}+ H (n = 2, l = 1)

p + Ps (n = 1)

p + Ps (n = 2, l = 0)

p + Ps (n = 2, l = 1)

- All Kmatrices are symmetric with an error less than 2%.
- All cross sections are small except that in resonant channels 5 and 6 that include all Hydrogen (antihydrogen) formation cross sections.

- c.
- Comparison with 2002 calculation [19].Reference [19] calculated a total of nine partial waves for a number of energies. The one that is closest to ${\mathrm{E}}_{1}$ is at 0.8749 Ry. According to Reference [19], the total hydrogen formation cross sections, including all nine partial waves, is 1670.02 $\pi {a}_{0}^{2}$. The contribution from the S-partial wave is 219.25 $\pi {a}_{0}^{2}$. The calculation from Reference [19] used the first generation of super computers, named Blue Horizon. The recent S-partial wave cross section at 0.8749 Ry is 276.77 $\pi {a}_{0}^{2}$. This calculation was carried out on a much more improved super computer, named Ranger. There is a significant increase in S-partial wave cross section. Accordingly, the total hydrogen formation cross section from the first resonance alone could be over 2000 $\pi {a}_{0}^{2}$.

## 4. Conclusions

- For the ${e}^{+}+H$ system, Reference [22] revealed only three S-state resonances above the $Ps\left(n=2\right)$ formation threshold. They are named Gailitis resonances due to their unique formation mechanism. The ${y}_{max}$ should be doubled to ~2000 a
_{0}to test the fine structure energy limit. Near the fine structure energy, the Coulomb degeneracy of the target atom is removed. Without the degeneracy, the incoming charged particle can not induce a first order electric dipole moment in the target atom [23]. Such Gailitis resonances are supported by a higher order Stark effect. - The MFE is able to provide wave amplitudes for each and every one of all the open channels. A simple three dimensional plot at a constant angle between $\stackrel{\rightharpoonup}{x}$ and $\stackrel{\rightharpoonup}{y}$ reveals important physics. For example [22], the structure of the wave amplitude along the x-axis reveals the bound states characteristics. Normally, along the y axis, one finds that the de Broglie wave structure, with the appropriate wave length, belongs to the channel plotted. If resonance exists in some channel or channels, wave packets appear along the y-axis. For the resonances listed in Table 2, Table 3 and Table 4 y
_{1}= 296.8 a_{0}, y_{2}= 702.4 a_{0}, and y_{3}= 1306 a_{0}. In this case, the physical “size” of the resonances are too large. In other cases [24], the energy widths are too wide. It will be a challenging task to find them using traditional methods, which are designed for the more compact portion of the Feshbach resonances. It is clear that the life-time of the wave packet formed along the y axis is the same as the life-time of the resonance. The width of the wave packet, ∆y, can be measured directly from the graph. The minimum uncertainty principle provides as good an estimate of the energy width as any other method. In Hu and Papp [24], the width of all 2nd order Stark-effect induced Gailitis resonances are obtained by searching the poles in the complex energy plane using the integral equation version of MFE. In addition, the positions of the wave packets y_{m}_{,}m is the quantum number of Gailitis resonances, which provided information to uncover the physical mechanism and Stark effect for Gailitis resonances. However, both Gailitis resonances, found above a threshold, and Feshbach resonances, found below a threshold, are induced by the same Coulomb field of the incoming charged particle. That is consistent with the Levinson theorem. Clearly, larger calculations capable of locating all resonances are necessary for both Feshbach and Gailitis resonances. - For the case investigated in Reference [22], the width of the wave packets measured from the plots can be approximated by the de Broglie wave length. Whether that can be generalized to high partial wave resonances must be determined with further calculations.

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Faddeev channel parameters [22], lengths in Bohr Radius.

Faddeev Channel α | ${\mathit{x}}_{\mathbf{0}}$ | ${\mathit{y}}_{\mathbf{0}}$ | ν | ${\mathit{y}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$ |
---|---|---|---|---|---|

One | 7.0 | 50.0 | 2.1 | 500.0 | 57.5 |

Two | 10.0 | 100.0 | 2.1 | 1000.0 | 81.0 |

${\mathit{E}}_{\mathbf{1}}$ = 0.8748; ${\mathit{\epsilon}}_{\mathbf{1}}$ = 0.34432 × 10^{−3} | |||||
---|---|---|---|---|---|

Cross Section Matrix | |||||

0.9998E−01 | 0.3191E−02 | 0.1913E−02 | 0.8951E−02 | 0.1399E−03 | 0.1611E−03 |

0.2230E−01 | 0.1046E+02 | 0.6338E+01 | 0.3918E+00 | 0.4924E+00 | 0.5676E+00 |

0.1336E−01 | 0.6338E+01 | 0.1258E+02 | 0.2602E+00 | 0.3274E+00 | 0.3774E+00 |

0.1044E−01 | 0.6538E−01 | 0.4343E−01 | 0.5063E+01 | 0.4397E−02 | 0.5068E−02 |

0.1778E+00 | 0.8958E+02 | 0.5957E+02 | 0.4794E+01 | 0.4710E+04 | 0.7676E+02 |

0.2048E+00 | 0.1032E+03 | 0.6866E+02 | 0.5524E+01 | 0.7676E+02 | 0.8756E+03 |

K-Matrix | |||||

−0.8299E−01 | 0.1070E+00 | −0.4787E+00 | 0.1557E+01 | −0.1148E+00 | −0.6303E−01 |

0.1070E+00 | 0.8278E+00 | −0.2225E+01 | 0.3038E+01 | −0.3513E+00 | −0.1929E+00 |

−0.4721E+00 | −0.2229E+01 | 0.3336E+01 | −0.9990E+01 | 0.1563E+01 | 0.8580E+00 |

0.1547E+01 | 0.3074E+01 | −0.1004E+02 | 0.3559E+02 | −0.2703E+01 | −0.1484E+01 |

−0.1154E+00 | −0.3560E+00 | 0.1563E+01 | −0.2682E+01 | 0.2990E+01 | 0.5194E+00 |

−0.6331E−01 | −0.1955E+00 | 0.8581E+00 | −0.1472E+01 | 0.5193E+00 | −0.3098E+00 |

${\mathit{E}}_{\mathbf{2}}$ = 0.87465; ${\mathit{\epsilon}}_{\mathbf{2}}$ = 0.19436 × 10^{−3} | |||||
---|---|---|---|---|---|

Cross Section Matrix | |||||

0.9996E−01 | 0.3311E−02 | 0.1907E−02 | 0.8921E−02 | 0.1300E−03 | 0.1630E−03 |

0.2315E−01 | 0.1030E+02 | 0.6402E+01 | 0.4000E+00 | 0.4559E+00 | 0.5709E+00 |

0.1334E−01 | 0.6402E+01 | 0.1255E+02 | 0.2693E+00 | 0.3031E+00 | 0.3795E+00 |

0.1040E−01 | 0.6670E−01 | 0.4490E−01 | 0.5060E+01 | 0.4054E−02 | 0.5080E−02 |

0.2927E+00 | 0.1468E+03 | 0.9755E+02 | 0.7826E+01 | 0.8987E+04 | 0.8021E+02 |

0.3669E+00 | 0.1838E+03 | 0.1222E+03 | 0.9807E+01 | 0.8021E+02 | 0.5349E+03 |

K-Matrix | |||||

−0.8962E−01 | 0.8594E−01 | −0.3864E+00 | 0.1402E+01 | 0.1345E+00 | −0.2820E−01 |

0.8617E−01 | 0.7618E+00 | −0.1940E+01 | 0.2553E+01 | 0.4102E+00 | −0.8601E−01 |

−0.3795E+00 | −0.1941E+01 | 0.2073E+01 | −0.7834E+01 | −0.1825E+01 | 0.3827E+00 |

0.1390E+01 | 0.2578E+01 | −0.7863E+01 | 0.3194E+02 | 0.3164E+01 | −0.6638E+00 |

0.1352E+00 | 0.4161E+00 | −0.1826E+01 | 0.3140E+01 | −0.3236E+01 | −0.4327E+00 |

−0.2840E−01 | −0.8727E−01 | 0.3829E+00 | −0.6592E+00 | −0.4327E+00 | 0.2335E+00 |

${\mathit{E}}_{\mathbf{3}}$ = 0.87454; ${\mathit{\epsilon}}_{\mathbf{3}}$ = 0.84344 × 10^{−4} | |||||
---|---|---|---|---|---|

Cross Section Matrix | |||||

0.1000E+00 | 0.2972E−02 | 0.1740E−02 | 0.8890E−02 | 0.2387E−03 | 0.2853E−03 |

0.2080E−011 | 0.9661E+01 | 0.5677E+01 | 0.3444E+00 | 0.8368E+00 | 0.1003E+01 |

0.1218E−01 | 0.5677E+01 | 0.1343E+02 | 0.2337E+00 | 0.5565E+00 | 0.6669E+00 |

0.1037E−01 | 0.5740E−01 | 0.3895E−01 | 0.5077E−01 | 0.7496E−02 | 0.8967E−02 |

0.1238E+01 | 0.6201E+03 | 0.4124E+03 | 0.3334E+02 | 0.1025E+04 | 0.2340E+03 |

0.1480E+01 | 0.7433E+03 | 0.4942E+03 | 0.3988E+02 | 0.2340E+03 | 0.1989E+05 |

K-Matrix | |||||

−0.6261E−01 | 0.1684E+00 | −0.7479E+00 | 0.2029E+01 | 0.3930E−01 | −0.4449E+00 |

0.1678E+00 | 0.1011E+01 | −0.3037E+01 | 0.4451E+01 | 0.1193E+00 | −0.1352E+01 |

−0.7426E+00 | −0.3053E+01 | 0.6953E+01 | −0.1628E+02 | −0.5306E+00 | 0.6011E+01 |

0.2026E+01 | 0.4518E+01 | −0.1637E+02 | 0.4669E+02 | 0.9248E+00 | −0.1047E+02 |

0.3956E−01 | 0.1212E+00 | −0.5312E+00 | 0.9195E+00 | −0.1623E+00 | 0.2459E+00 |

−0.4471E+00 | −0.1373E+01 | 0.6016E+01 | −0.1040E+02 | 0.2460E+00 | 0.7620E+01 |

Ch | 1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|---|

Ch | |||||||

1 | ${e}^{+}+H\left(n\right)\to {e}^{+}+H\left({n}^{\prime}\right)$ | ${e}^{+}+H\left(n\right)\to p+Ps\left({n}^{\prime}\right)$ | |||||

2 | $n=1,\text{}2,\text{}3$ | $n=1,\text{}2,\text{}3$ | |||||

3 | ${n}^{\prime}=1,\text{}2,\text{}3$ | ${n}^{\prime}=4,\text{}5,\text{}6$ | |||||

4 | $p+Ps\left(n\right)\to {e}^{+}+H\left({n}^{\prime}\right)$ | $p+Ps\left(n\right)\to p+Ps\left({n}^{\prime}\right)$ | |||||

5 | $n=4,\text{}5,\text{}6$ | $n=4,\text{}5,\text{}6$ | |||||

6 | ${n}^{\prime}=1,\text{}2,\text{}3$ | ${n}^{\prime}=4,\text{}5,\text{}6$ |

${\mathit{\sigma}}_{{\mathit{\epsilon}}_{\mathbf{1}}}$ | ${\mathit{\sigma}}_{{\mathit{\epsilon}}_{\mathbf{2}}}$ | ${\mathit{\sigma}}_{{\mathit{\epsilon}}_{\mathbf{3}}}$ |
---|---|---|

321.51 | 551.13 | 2272.82 |

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Hu, C.Y. The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems. *Atoms* **2016**, *4*, 16.
https://doi.org/10.3390/atoms4020016

**AMA Style**

Hu CY. The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems. *Atoms*. 2016; 4(2):16.
https://doi.org/10.3390/atoms4020016

**Chicago/Turabian Style**

Hu, Chi Yu. 2016. "The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems" *Atoms* 4, no. 2: 16.
https://doi.org/10.3390/atoms4020016