Four-body problem in d-dimensional space: ground state, (quasi)-exact-solvability. IV

Abstract

Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a $4$-body system with equal masses in {\it $d$}-dimensional space 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. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For $d\ge 3$, this describes a six-dimensional quantum particle moving in a curved space with special $d$-independent metric in a certain $d$-dependent singular potential, while for $d=1$ it corresponds to a three-dimensional particle and coincides with the $A_3$ (4-body) rational Calogero model; the case $d=2$ is exceptional and is discussed separately. The kinetic energy of the system has a hidden $sl(7,\mathbf{R})$ Lie (Poisson) algebra structure, but for the special case $d=1$ it becomes degenerate with hidden algebra $sl(4,R)$. We find an exactly-solvable four-body $S_4$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable four-body sextic polynomial type potential with singular terms. Naturally, the tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, that lead to exactly solvable models. Their generalization to the $n$-body system as well as the case of non-equal masses is briefly discussed.

arXiv comment42 pages, 1 figure

References in corpus (8)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Exact Solvability of the Calogero and Sutherland Models
• Quasi-Exactly-Solvable Many-Body Problems
• Hidden Algebra of Three-Body Integrable Systems
• Three-body problem in 3D space: ground state, (quasi)-exact-solvability
• Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability
• The quantum n-body problem in dimension $d\ge n-1$: ground state
• Quantum four-body system in D dimensions

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• Classical $n$-body system in volume variables. II. Four-body case
• On the four-body problem in the Born-Oppenheimer approximation