Exact analytical solution of the Schroedinger equation for an N-identical body-force system

A mathematical method is presented for solving the Schroedinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated with the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ12 + ξ22 + ... + ξN-12)1/2 only, where ξ1,ξ2, ..., ξN-1 are Jacobi relative coordinates which are functions of N-particle relative positions r12, r23, ..., rN1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax2 + bx - (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schroedinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles. (author)