Exactly Solvable $N$ -Body Quantum Systems with $N=3^k (k$ $\ge $ $2)$ in the $D=1$ Dimensional Space

We study the exact solutions of a particular class of N confined particles of equal mass, with N = 3(k) (k = 2, 3,...), in the D = 1 dimensional space. The particles are clustered in clusters of three particles. The interactions involve a confining mean field, two-body Calogero type of potentials inside the cluster, interactions between the centres of mass of the clusters and finally a non-translationally invariant N-body potential. The case of nine particles is exactly solved, in a first step, by providing the full eigensolutions and eigenenergies. Extending this procedure, the general case of N particles (N = 3(k), k >= 2) is studied in a second step. The exact solutions are obtained via appropriate coordinate transformations and separation of variables. The eigenwave functions and the corresponding energy spectrum are provided.