2017/07/31 by Alexander V. Turbiner, Alexander V Turbiner, Willard Miller +2
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Configuration space #Eigenfunction #Generalization #Ground state #Hamiltonian (control theory) #Harmonic oscillator #Lie algebra #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum potential #Quantum system #math-ph #math.MP #nlin.SI #quant-ph
paper · pdf · doi:10.1063/1.4994397
published as Journal of Mathematical Physics A59 (2018) 022108 (29pp) · 49 pages, 1 figure, 27 references, generalization and extension of arXiv:1611.08157, further extension of Version 1, a notion of geometrical variables introduced and (quasi)-exactly-solvable problem in these variables described
openalex created_date 2017/07/14 · arxiv created 2018/02/01 · openalex publication_date 2018/02/01 · arxiv updated 2018/03/01 · openalex updated_date 2026/08/05
As a straightforward generalization and extension of our previous paper [A. V. Turbiner et al., “Three-body problem in 3D space: Ground state, (quasi)-exact-solvability,” J. Phys. A: Math. Theor. 50, 215201 (2017)], we study the aspects of the quantum and classical dynamics of a 3-body system with equal masses, each body with d degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. The quantum (and classical) Hamiltonian for which these states are eigenfunctions is derived. It corresponds to a three-dimensional quantum particle moving in a curved space with special d-dimension-independent metric in a certain d-dependent singular potential, while at d = 1, it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is d-independent; it has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h(3) typical for the H3 Calogero model as in the d = 3 case. We find an exactly solvable three-body S3-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly solvable three-body sextic polynomial type potential with singular terms. For both models, an extra first order integral exists. For d = 1, the whole family of 3-body (two-dimensional) Calogero-Moser-Sutherland systems as well as the Tremblay-Turbiner-Winternitz model is reproduced. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to d > 1 leads to two primitive quasi-exactly solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.