  $N$-body non efimovian bound or quasi-bound states for particles with short
range interactions are considered in arbitrary dimensions. The different
resonance regimes near the threshold are depicted by using a generalization of
the effective range approximation. This two-parameter description can be used
in various contexts from ultra-cold to hadronic physics. The universal
character of these states makes it possible a formulation in terms of a contact
model. The singularity at the contact imposes the introduction of a modified
scalar product to solve the normalization catastrophe and to restore the
self-adjoint character of the model. An equivalence with the standard scalar
product used for realistic finite range models is derived.
