  It is shown that the scattering amplitude for contact-interacting anyons does
exhibit a genuine non-perturbative sector. This means that the corresponding
perturbative field theoretical formulation, based on the {\it 2+1}
non-relativistic Chern-Simons gauge model coupled to self-interacting complex
scalar field, is not generally able to reproduce, order by order in
perturbation theory, the exact result. It is proven that the full agreement
between the exact scattering amplitude and the resummation of the perturbative
expansion of the renormalized 1PI amplitude actually occurs only for some
continuous sub-family of self-adjoint extensions of the quantum Hamiltonians,
which entail the absence of bound states. A comparison with previously obtained
results is carefully worked out.
