  We show that the N-particle A_{N-1} and B_N rational Calogero models without
the harmonic interaction admit a new class of bound and scattering states.
These states owe their existence to the self-adjoint extensions of the
corresponding Hamiltonians, labelled by a real parameter z. It is shown that
the new states appear for all values of N and for specific ranges of the
coupling constants. Moreover, they are shown to exist even in the excited
sectors of the Calogero models. The self-adjoint extension generically breaks
the classical scaling symmetry, leading to quantum mechanical scaling anomaly.
The scaling symmetry can however be restored for certain values of the
parameter z. We also generalize these results for many particle systems with
classically scale invariant long range interactions in arbitrary dimensions.
