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**Diffraction approach in the scattering problem for three charged quantum particles.**
*(English.
Russian original)*
Zbl 1451.81368

Math. Notes 108, No. 3, 457-461 (2020); translation from Mat. Zametki 108, No. 3, 469-473 (2020).

From the introduction: The results obtained in this paper are based on the diffraction approach in the theory of scattering. The approach was proposed by V. S, Buslaev, S. P. Merkur’ev and S. P. Salikov [J. Sov. Math. 21, 260–265 (1983; Zbl 0515.35069)] to describe the asymptotics of the solution at infinity of the problem of scattering of three short-range one-dimensional quantum particles. We also note the paper [M. Gaudin et al., J. Phys. 36, No. 12, 1183–1197 (1975; doi:10.1051/jphys:0197500360120117700)]. Later, it was also shown by the author [J. Math. Sci., New York 226, No. 6, 744–767 (2017; Zbl 1380.81121)] that the proposed method can be generalized to the case of slow (Coulomb) decrease in pair potentials of repulsion in systems of both one-dimensional and multidimensional particles. In particular, a generalization of results of [E. O. Alt and A. M. Mukhamedzhanov, JETP Lett. 56, No. 9, 435 (1992)] was obtained.

We note that the situation becomes most complicated and interesting in the case of the scattering problem of three three-dimensional Coulomb particles in the presence of pair potentials of attraction [S. P. Merkur’ev and L. D. Faddeev, Квантовая теория рассеяния для систем нескольких частиц (Russian). Moskva: “Nauka”, Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 400 p. (1985; Zbl 0585.35078)]. A realization of the scheme developed in the framework of diffraction approach, which was described in [the author et al., J. Math. Sci., New York 238, No. 5, 601–620 (2019; Zbl 1419.81038)], allows us to formulate Theorem 1.

We note that such a contribution generated by different pair subsystems (with Coulomb pair potentials of attraction) must be determined independently. Such contributions become substantial in various asymptotic domains of the configuration space when Coulomb scattering is described.

In what follows, we show how an asymptotics of the form (1) can be used to describe quantum scattering with infinitely many asymptotic channels. We consider the scattering problem for three three-dimensional quantum charged particles in the presence of Coulomb pair potentials of attraction. For definiteness, we assume that there is initially a charged particle and a two-particle cluster in the Coulomb bound state. We also assume that the total energy of the system admits the decay of the system, i.e., admits \(2 \rightarrow 3\) processes.

Our goal is to determine the complete set of scattering amplitudes corresponding to different processes.

We note that the situation becomes most complicated and interesting in the case of the scattering problem of three three-dimensional Coulomb particles in the presence of pair potentials of attraction [S. P. Merkur’ev and L. D. Faddeev, Квантовая теория рассеяния для систем нескольких частиц (Russian). Moskva: “Nauka”, Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 400 p. (1985; Zbl 0585.35078)]. A realization of the scheme developed in the framework of diffraction approach, which was described in [the author et al., J. Math. Sci., New York 238, No. 5, 601–620 (2019; Zbl 1419.81038)], allows us to formulate Theorem 1.

We note that such a contribution generated by different pair subsystems (with Coulomb pair potentials of attraction) must be determined independently. Such contributions become substantial in various asymptotic domains of the configuration space when Coulomb scattering is described.

In what follows, we show how an asymptotics of the form (1) can be used to describe quantum scattering with infinitely many asymptotic channels. We consider the scattering problem for three three-dimensional quantum charged particles in the presence of Coulomb pair potentials of attraction. For definiteness, we assume that there is initially a charged particle and a two-particle cluster in the Coulomb bound state. We also assume that the total energy of the system admits the decay of the system, i.e., admits \(2 \rightarrow 3\) processes.

Our goal is to determine the complete set of scattering amplitudes corresponding to different processes.

### MSC:

81U10 | \(n\)-body potential quantum scattering theory |

81V10 | Electromagnetic interaction; quantum electrodynamics |

81U35 | Inelastic and multichannel quantum scattering |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

47A40 | Scattering theory of linear operators |

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\textit{S. B. Levin}, Math. Notes 108, No. 3, 457--461 (2020; Zbl 1451.81368); translation from Mat. Zametki 108, No. 3, 469--473 (2020)

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### References:

[1] | Buslaev, V. S.; Merkur’ev, S. P.; Salikov, S. P., Problems of Mathematical Physics, Vol. 9: Scattering Theory. Theory of Oscillations, 14-30 (1979), Leningrad: Izd. Leningrad Univ., Leningrad · Zbl 0494.47010 |

[2] | Buslaev, V. S.; Merkur’ev, S. P.; Salikov, S. P., J. Sov. Math., 21, 3, 260 (1983) · Zbl 0515.35069 |

[3] | Gaudin, M.; Derrida, B., J. Phys., 36, 1183 (1975) |

[4] | Buslaev, V. S.; Levin, S. B., St. Petersburg Math. J., 22, 3, 379 (2010) · Zbl 1219.81235 |

[5] | Buslaev, V. S.; Levin, S. B., Functional Anal. Appl., 46, 2, 147 (2012) · Zbl 1272.81185 |

[6] | Levin, S. B., J. Math. Sci. (N.Y.), 226, 6, 744 (2017) · Zbl 1380.81121 |

[7] | Alt, E. O.; Mukhamedzhanov, A. M., JETP Lett., 56, 9, 435 (1992) |

[8] | Merkur’ev, S. P.; Faddeev, L. D., Quantum Scattering Theory for Systems of Several Particles (1985), Moscow: Nauka, Moscow · Zbl 0585.35078 |

[9] | Larsson, M.; Orel, A. E., Dissociative Recombination of Molecular Ions (2008), Cambridge: Cambridge Univ. Press, Cambridge |

[10] | Tennyson, J., Phys. Rep., 491, 29 (2010) |

[11] | Budylin, A. M.; Koptelov, Ya. Yu.; Levin, S. B., J. Math. Sci. (New York), 238, 5, 601 (2019) · Zbl 1419.81038 |

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