Three-body problem in 3D space: Ground state, (quasi)-exact-solvability

Alexander V. Turbiner; Willard Miller; Adrian M. Escobar-Ruiz

doi:10.1088/1751-8121/aa6cc2

Three-body problem in 3D space: Ground state, (quasi)-exact-solvability

Alexander V. Turbiner, Willard Miller, Adrian M. Escobar-Ruiz

School of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We study aspects of the quantum and classical dynamics of a 3-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h ⁽³⁾ typical for the H ₃ Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.

Original language	English (US)
Article number	215201
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	50
Issue number	21
DOIs	https://doi.org/10.1088/1751-8121/aa6cc2
State	Published - May 2 2017

Bibliographical note

Funding Information:
WM was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller, Jr). MAE is grateful to ICN UNAM, Mexico for the kind hospitality during his visit, where a part of the research was done, he was supported in part by DGAPA grant IN108815 (Mexico) and, in general, by CONACyT grant 250881 (Mexico) for postdoctoral research.

Publisher Copyright:
© 2017 IOP Publishing Ltd.

Keywords

(quasi)-exact-solvability
hidden algebra
three-body problem

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10.1088/1751-8121/aa6cc2

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abstract = "We study aspects of the quantum and classical dynamics of a 3-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h (3) typical for the H 3 Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.",

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N2 - We study aspects of the quantum and classical dynamics of a 3-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h (3) typical for the H 3 Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.

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Three-body problem in 3D space: Ground state, (quasi)-exact-solvability

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