Four-body (an)harmonic oscillator in $d$-dimensional space: $S$-states, (quasi)-exact-solvability, hidden algebra $sl(7)$

Abstract

As a generalization and extension of our previous paper {\it J. Phys. A: Math. Theor. 53 055302} \cite{AME2020}, in this work we study a quantum 4-body system in ${\mathbb{R}}^d$ ($d\ge 3$) with quadratic and sextic pairwise potentials in the {\it relative distances}, $r_{ij}\equiv |{\mathbf{r}}_i-{\mathbf{r}}_j|$, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum ($S$-states). In variables ${\rho }_{ij}\equiv r_{ij}^2$, the corresponding reduced Hamiltonian of the system possesses a hidden $sl(7;\mathbf{R})$ Lie algebra structure. In the $\rho$-representation it is shown that the 4-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly-solvable (ES). We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite, three others are equal), molecular two-center (two masses are infinite, two others are equal) and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born-Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. Also, the reduction to the lower dimensional cases $d=1,2$ is discussed. It is shown that for four body harmonic oscillator case there exists an infinite family of eigenfunctions which depend on the single variable which is the moment-of-inertia of the system.

arXiv comment39 pages, 2 figures

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

Cited by in corpus (4)

• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Two-body neutral Coulomb system in a magnetic field at rest: from Hydrogen atom to positronium
• Classical $n$-body system in volume variables. II. Four-body case
• On the four-body problem in the Born-Oppenheimer approximation