Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability

Abstract

As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 we study 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. 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 $H_3$ Calogero model as in the $d=3$ case. We find an exactly-solvable three-body $S^3$-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. 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.

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

References in corpus (1)

• From Quantum $A_N$ to $E_8$ Trigonometric Model: Space-of-Orbits View

Cited by in corpus (11)

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• The $n$ body matrix and its determinant
• Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$
• Four-body (an)harmonic oscillator in $d$-dimensional space: $S$-states, (quasi)-exact-solvability, hidden algebra $sl(7)$
• Four-body problem in d-dimensional space: ground state, (quasi)-exact-solvability. IV
• From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics
• Classical $n$-body system in geometrical and volume variables. I. Three-body case
• Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions
• 3-body harmonic molecule
• Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)