Generalized three-body harmonic oscillator system: ground state

Adrian M. Escobar-Ruiz; Fidel Montoya

doi:10.14311/AP.2022.62.0050

Authors

Adrian M. Escobar-Ruiz Universidad Autónoma Metropolitana, Departamento de Física, Av. San Rafael Atlixco 186, 09340 Ciudad de México, CDMX, México
Fidel Montoya Universidad Autónoma Metropolitana, Departamento de Física, Av. San Rafael Atlixco 186, 09340 Ciudad de México, CDMX, México

DOI:

https://doi.org/10.14311/AP.2022.62.0050

Keywords:

three-body system, exact-solvability, hidden algebra, integrability

Abstract

In this work we report on a 3-body system in a d−dimensional space ℝ^d with a quadratic harmonic potential in the relative distances rij = |ri −rj| between particles. Our study considers unequal masses, different spring constants and it is defined in the three-dimensional (sub)space of solutions characterized (globally) by zero total angular momentum. This system is exactly-solvable with hidden algebra sℓ₄(ℝ). It is shown that in some particular cases the system becomes maximally (minimally) superintegrable. We pay special attention to a physically relevant generalization of the model where eventually the integrability is lost. In particular, the ground state and the first excited state are determined within a perturbative framework.

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