Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol