  We discuss the problem of constructing self-adjoint and lower bounded
Hamiltonians for a system of $n>2$ non-relativistic quantum particles in
dimension three with contact (or zero-range or $\delta$) interactions. Such
interactions are described by (singular) boundary conditions satisfied at the
coincidence hyperplanes, i.e., when the coordinates of two particles coincide.
Following the line of recent works appeared in the literature, we introduce a
boundary condition slightly modified with respect to usual boundary condition
one has in the one-body problem. With such new boundary condition we can show
that the instability property due to the fall to the center phenomenon
described by Minlos and Faddeev in 1962 is avoided. Then one obtains a
physically reasonable Hamiltonian for the system. We apply the method to the
case of a gas of $N$ interacting bosons and to the case of $N$ distinguishable
particles of equal mass $M$ interacting with a different particle. In the
latter case we also discuss the limit of the model for $M \longrightarrow
+\infty$. We show that in the limit one obtains the one-body Hamiltonian for
the light particle subject to $N$ (non-local) point interactions placed at
fixed positions. We will verify that such non-local point interactions do not
exhibit the ultraviolet pathologies that are present in the case of standard
local point interactions.
