Zero range interactions in d=3 and d=2 revisited

This paper has a two-fold purpose: 1) to clarify the difference between contact and weak-contact interactions (called point interactions in [A] in the case $N=2$) in three dimensions and their role in providing spectral properties and boundary conditions. 2) to analyze the same problem in two dimensions. Both contact and weak-contact are "zero range interactions" or equivalently self-adjoint extension of the symmetric operator $\hat H_0$, the free hamiltonian for a system of $N$ particles, restricted to functions that vanish in some neighborhood of the \emph{contact manyfold} $\Gamma = \cup_{i,j} \{ x_i - x_j = 0\} \;\; i \not= j =1 \ldots N$. The \emph{hamiltonian formulation} of a weak-contact interaction requires the presence of a zero energy resonance. Both can be obtained, for $N \geq 3$, as scaling limit, in the strong resolvent sense, of hamiltonians with two-body central potentials of very short range; the scaling is different in the two cases. The contact "potential" is a distribution on the "contact manyfold"; the laplacian is now two dimensional and the scaling properties under dilation of the laplacian are now the same as those of the contact interaction. This makes a major difference in the spectral properties. Contact interaction defines now a hamiltonian system. In two dimensions one can consider a hamiltonian system of three particle of mass $m$ having as potential a contact potential and \emph{the product of two weak-contact potentials}. In the limit $m \to \infty$ their wave functions have vanishingly small support and the system may be regarded as a \emph{point with internal structure}; the spectrum has an essential singularity at the bottom of the continuous part. The Wave Operator for the interaction with a fourth particle extends to a bounded map on $L ^p$ for all $1 < p < \infty$ [E,G,G]