  We investigate the existence of bound states in a one-dimensional quantum
system of $N$ identical particles interacting with each other through an
inverse square potential. This system is equivalent to the Calogero model
without the confining term. The effective Hamiltonian of this system in the
radial direction admits a one-parameter family of self-adjoint extensions and
the negative energy bound states occur when most general boundary conditions
are considered. We find that these bound states exist only when $N=3,4$ and for
certain values of the system parameters. The effective Hamiltonian for the
system is related to the Virasoro algebra and the bound state wavefunctions
exhibit a scaling behaviour in the limit of small inter-particle separation.
