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On justification of the asymptotics of eigenfunctions of the absolutely continuous spectrum in the problem of three one-dimensional short-range quantum particles with repulsion. (English. Russian original) Zbl 1419.81009

J. Math. Sci., New York 238, No. 5, 566-590 (2019); translation from Zap. Nauchn. Semin. POMI 461, 14-51 (2017).
Summary: The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrödinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of \(N\) particles with arbitrary masses is discussed.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
47A10 Spectrum, resolvent
35B40 Asymptotic behavior of solutions to PDEs
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[1] Budylin, AM; Buslaev, VS, Reflection operator and their applications to asymptotic investigations of semiclassical integral equations, Adv. Soviet Math, 7, 107-157, (1991) · Zbl 0743.45003
[2] Budylin, AM; Buslaev, VS, Semiclassical asymptotics of the resolvent of the integral convolution operator with the sine-kernel on a finite interval, St. Petersburg Math. J., 7, 925-942, (1996)
[3] Mourre, E., Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys., 78, 391-408, (1981) · Zbl 0489.47010
[4] K. Mauren, Methods of the Hilbert Space [Russian translation], Mir (1965).
[5] L. D. Faddeev, Mathematical Aspects of the Three-Body Problem of the Quantum Scattering Theory, Daniel Davey and Co., Inc. (1965).
[6] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators, AP (1978).
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III, Scattering Theory, AP (1979).
[8] Perry, P.; Sigal, IM; Simon, B., Spectral analysis of N-body Schrödinger operators, Ann. Math., 114, 519-567, (1981) · Zbl 0477.35069
[9] Buslaev, VS; Levin, SB; Neittaanmäki, P.; Ojala, T., New approach to numerical computation of the eigenfunctions of the continuous spectrum of the three-particle Schrödinger operator: I. One-dimensional particles, short-range pair potentials, J. Phys. A: Math.Theor., 43, 285-205, (2010) · Zbl 1193.81111
[10] Buslaev, VS; Levin, SB, Asymptotic behavior of the eigenfunctions of the manyparticle Schrödinger operator. I. One-dimensional particles, Amer. Math. Soc. Transl., 225, 55-71, (2008) · Zbl 1160.81476
[11] Buslaev, VS; Merkuriev, SP; Salikov, SP, On diffractional character of scattering in a quantum system of three one-dimensional particles, Probl. Mat. Fiz., Leningrad. Univ., Leningrad, 9, 14-30, (1979)
[12] Buslaev, VS; Merkuriev, SP; Salikov, SP, Description of pair potentials for which the scattering in a quantum system of three one-dimensional particles is free of diffraction effects, Zap. Nauchn. Semin. LOMI, 84, 16-22, (1979) · Zbl 0413.35058
[13] D. R. Yafaev, Mathematical Scattering Theory, Americ. Math. Soc. (1992).
[14] I. M. Gelfand and N. Ya. Vilenkin, Some Applications of Harmonic Analysis. Framed Hilbert Spaces (Generalized Functions) [in Russian], FM (1961).
[15] L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particle Systems, Kluwer, Dordrecht (1993). · Zbl 0797.47005
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