Exactly solvable $N$-body quantum systems with $N=3^k \ ( k \geq 2)$ 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 3 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 9 particles is exactly solved, in a first step, by providing the full eigensolutions and eigenenergies. Extending this procedure, the general case of $N$ particles ($N=3^k, \ k \geq 2$) 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.