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AIP conference proceedings

Uniform asymptotics of eigenfunctions for the three-body Schrodinger operator in one-dimensional case

V. S. Buslaev, S. B. Levin, Jacob Schach Møller, Dmitri Fedorov

Published Jan 1, 2008• 0 citations

Abstract

The three‐body scattering problem with finite pair potentials for one‐dimensional case is investigated. The asymptotic function χ0, which satisfies the three‐body Schrödinger equation in whole configuration space outside of compact domain Ω, where the supports of all three pair potentials cross each other, has been presented in a mathematically rigorous way. For large distances |x|→∞ the function χ0 determines the asymptotics of the solution up to the circular wave with smooth coefficient in whole configuration space. The method is based on analogies between few‐body scattering problem and diffraction one of the plane wave on the system of half‐transparent infinite screens. Presented here formalism are believed to be useful also for the few‐body scattering problem of higher dimensions.

DOI: 10.1063/1.2915630

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The three‐body scattering problem with finite pair potentials for one‐dimensional case is investigated. The asymptotic function χ0, which satisfies the three‐body Schrödinger equation in whole configuration space outside of compact domain Ω, where the supports of all three pair potentials cross each other, has been presented in a mathematically rigorous way. For large distances |x|→∞ the function χ0 determines the asymptotics of the solution up to the circular wave with smooth coefficient in whole configuration space. The method is based on analogies between few‐body scattering problem and diffraction one of the plane wave on the system of half‐transparent infinite screens. Presented here formalism are believed to be useful also for the few‐body scattering problem of higher dimensions.

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.

On the leading term of asymptotics of the

n

like-charged quantum particles scattering problem solution

An ansatz describing in terms of formal asymptotic decompositions a leading term of asymptotics of the

n

three-dimensional like-charged quantum particles scattering problem solution is suggested. The description of the solution in those asymptotic configurations in which it was known earlier (for example,

n=3

), coincides with the previously known constructions \cite{BBK,z1,am92,BL2,KL}. It is shown that the Schredinger equation discrepancy for the suggested ansatz decreases faster than the potential uniformly in all angle variables at infinity in configuration space. An assumption is made about the structure of the leading term of the asymptotics of the scattering problem solution related to the

n

three-dimensional quantum particles interacting by a broad class of slowly decreasing pair potentials.

The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.