The present paper aims at announcing a new approach to the construction of the asymptotics (at infinity in configuration space) of the resolvent kernel of the Schrödinger operator in the scattering problem of three one-dimensional quantum particles interacting by compactly supported pair repulsive potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator can be constructed explicitly. It should be emphasized that the restriction of the consideration to the case of compactly supported pair potentials does not lead to a simplification of the problem in its essence, since the potential of the interaction of all three particles remains nondecreasing at infinity, but allows one to put aside a certain number of technical details.
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References
A. M. Budylin and V. S. Buslaev, “Reflection operators and their applications to asymptotic investigations of semiclassical integral equations,” Adv. Sov. Math., Amer. Math. Soc., 7, No. 6, 79–103 (1991).
L. D. Faddeev, “Mathematical aspects of the three-body problem of the quantum scattering theory,” Daniel Davey and Co., Inc. (1965).
V. S. Buslaev, S. B. Levin, P. Neittaanmäki, and T. Ojala, “New approach to numerical computation of the eigenfunctions of the continuous spectrum of three-particle Schrödinger operator: I. One-dimensional particles, short-range pair potentials,” J. Phys. A, 43, 285205 (2010).
V. S. Buslaev and S. B. Levin, “Asymptotic Behavior of the Eigenfunctions of the Manyparticle Shrödinger Operator. I. One-dimentional Particles,” Amer. Math. Soc. Transl., 2, 225, 55–71 (2008).
A. M. Budylin and V. S. Buslaev, “The quasiclassical asymptotics of the resolvent of the integral convolution operator with sine kernel on a finite interval,” Adv. Sov. Math., Amer. Math. Soc., 7, 107–157 (1995).
E. Mourre, “Absence of singular continuous spectrum for certain self-adjoint operators,” Commun. Math. Phys., 78, 391–408 (1981).
P. Perry, I. M. Sigal, and B. Simon, “Spectral analysis of N-body Schrodinger operators,” Annals of Mathematics, 114, 519–567 (1981).
D. R. Yafaev, Mathematical Scattering Theory: Analytic Theory, Math. Surveys and Monographs, 158 (2010).
I. M. Gel’fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4, Academic Press (1964).
K. Moren, Methods of Hilbert Spaces [Russian translation], Mir, Moscow (1965).
M. Reed and B. Simon, Methods of Modern Mathematical Physics [Russian translation], Vol. 3, Mir, Moscow (1982).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 438, 2015, pp. 95–103.
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Budylin, A.M., Levin, S.B. To the Question on the Resolvent Kernel Asymptotics in the Three-Body Scattering Problem. J Math Sci 224, 63–68 (2017). https://doi.org/10.1007/s10958-017-3394-4
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DOI: https://doi.org/10.1007/s10958-017-3394-4