Three-body closed chain of interactive (an)harmonic oscillators and the algebra

Abstract

In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, , between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator with hidden algebra , while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only (S-states). We pay special attention to the cases where all masses of the particles and spring constants are unequal as well as to the atomic, where one mass is infinite, and molecular, where two masses are infinite, limits; it is a non-integrable system. In general, a three-body harmonic oscillator is 7-parametric depending on three masses and three spring constants, and frequency; it is an exactly-solvable problem with spectra linear in three quantum numbers and with hidden algebra . In particular, the first and second order integrals of the 3-body oscillator for unequal masses are searched: it is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.