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**The scattering problem of three one-dimensional short-range quantum particles involving bound states in pair subsystems. The coordinate asymptotics of the resolvent kernel and of the eigenfunctions of the absolutely continuous spectrum.**
*(English.
Russian original)*
Zbl 1455.81044

J. Math. Sci., New York 252, No. 5, 567-575 (2021); translation from Zap. Nauchn. Semin. POMI 483, 5-18 (2019).

Summary: In the present work, we consider the scattering problem of three one-dimensional quantum particles of equal mass interacting by pair finite potentials such that each pair subsystem permits a bound state. We study the limit values of the Schrödinger operator resolvent integral kernel as the spectral parameter approaches the positive semiaxis, which allows us to construct the asymptotics of eigenfunctions of the absolutely continuous spectrum.

### MSC:

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

81V45 | Atomic physics |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |

47A10 | Spectrum, resolvent |

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\textit{I. V. Baibulov} et al., J. Math. Sci., New York 252, No. 5, 567--575 (2021; Zbl 1455.81044); translation from Zap. Nauchn. Semin. POMI 483, 5--18 (2019)

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### References:

[1] | Budylin, AM; Buslaev, VS, Reflection operator and their applications to asymptotic investigations of semiclassical integral equations, Advances in Soviet Math., AMS, Providence, RI, 7, 107-157 (1991) · Zbl 0743.45003 |

[2] | Baibulov, IV; Budylin, MA; Levin, SB, The asymptotics of eigenfunctions of the absolutely continuous spectrum. The scattering problem of three one-dimensional quantum particles, J. Math. Sci., 243, 5, 640-655 (2019) · Zbl 1435.81230 |

[3] | Baibulov, IV; Budylin, MA; Levin, SB, On justification of the asymptotics of eigenfunctions of the absolutely continuous spectrum in the problem of three onedimensional short-range quantum particles with repulsion, J. Math. Sci., 238, 5, 566-590 (2019) · Zbl 1419.81009 |

[4] | Faddeev, LD, Mathematical aspects of the three-body problem of the quantum scattering theory (1965), Inc., Jerusalem: Daniel Davey and Co., Inc., Jerusalem · Zbl 0131.43504 |

[5] | Mourre, E., Absence of a singular continuous spectrum for certain self-adjoint operators, Commun. Math.Phys., 78, 391-408 (1991) · Zbl 0489.47010 |

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