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arXiv:0909.4529 · Journal of Physics A Mathematical and Theoretical

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

V. S. Buslaev, S. B. Levin, Pekka Neittaanmäki, Tuomo Ojala

Published Jun 16, 2010• 14 citations
arXivHTMLPDF

Abstract

Basing on analogy between the three-body scattering problem and the diffraction problem of the plane wave (for the case of the short range pair potentials) by the system of six half transparent screens, we presented a new approach to the few-body scattering problem. The numerical results have been obtained for the case of the short range nonnegative pair potentials. The presented method allows a natural generalization to the case of the long range pair potentials.

DOI
10.1088/1751-8113/43/28/285205

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Basing on analogy between the three-body scattering problem and the diffraction problem of the plane wave (for the case of the short range pair potentials) by the system of six half transparent screens, we presented a new approach to the few-body scattering problem. The numerical results have been obtained for the case of the short range nonnegative pair potentials. The presented method allows a natural generalization to the case of the long range pair potentials.
We consider the quantum scattering problem of three one-dimensional particles with repulsive shortrange pair potentials. For clarity, we restrict ourself by the case of finite pair potentials. The absence of singular continuous spectrum of the corresponding Schröedinger operator for a broad class of pair potentials was proved earlier in a known work of E. Mourre. Nevertheless, the Mourre techniques do not allow description of the asymptotics of absolutely continuous spectrum eigenfunctions. In this work, regardless of Mourre results, we prove the existence of the resolvent limit values on absolutely continuous spectrum and construct them explicitly. It allows us, following the known procedure, to derive the absolutely continuous spectrum eigen-functions asymptotics, suggested earlier in works of V. S. Buslaev and his co-authors. Our approach, close to the foundational work of L. D. Faddeev devoted to three-dimensional particles, specifically uses the ideas of Schwarz alternating method.
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