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Journal of Experimental and Theoretical Physics

Quantum Scattering of the Bound Pair on the Third Particle in One-Dimensional Case

Alexander Mikhajlovich Budylin, S. B. Levin

Published Nov 1, 2022• 0 citations
Publisher

Abstract

Abstract In this paper, the problem of scattering 2 → 2(3) of three 1D quantum particles with pair short-range attraction potentials is considered in the framework of the diffraction approach. The solution to the scattering problem is constructed in terms of the solution to the model nonhomogeneous boundary-value problem in a circle of a large radius with radiation conditions on the boundary. Possible physical applications of the constructed model are studied.

DOI
10.1134/s1063776122110152

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Abstract In this paper, the problem of scattering 2 → 2(3) of three 1D quantum particles with pair short-range attraction potentials is considered in the framework of the diffraction approach. The solution to the scattering problem is constructed in terms of the solution to the model nonhomogeneous boundary-value problem in a circle of a large radius with radiation conditions on the boundary. Possible physical applications of the constructed model are studied.
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.
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.