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\begin{document}

\title{Contact interactions via Gamma convergence }


We study for a $N$-body system in $R^3$ (and $R^2$)  \emph{contact interactions} i.e. interactions supported by the coincidence sets $ \{x_i = x_j\} $; they are described by boundary conditions or alternatively by potentials which are distributions supported by the coincidence sets. 

Contact interactions may be of \emph{strong} or \emph{weak} type. Weak contact is accompanied by a zero energy resonance. Both are  self-adjoint extensions of the free hamiltonian restricted away from  the coincidence sets. If one of the particles has infinite mass , weak contact of two bodies is called  \emph{point interaction}. We will prove that strong and weak contact hamiltonians produce \emph{independent} and \emph{complementary} effects. 

In three dimensions strong contact interactions are responsible for the Efimov spectrum  of three and four body bound states in high energy physics. Weak contact is responsible for the Bose Einstein condensate (a gas of three particles weakly bound that satisfy  the Gross-Pitaewsii equation). In an Appendix we prove that weak contact  describe also the structure of the Polaron in Nelson's model of a particle weakly coupled to a mass zero quantized field. 

We  consider briefly the two dimensional case.  Weak contact is not present; Bose-Einstein condensate\emph{is due to a very special long range interaction}. 

In the one-dimensional case (a graph) there are again two types of contact interactions. If the particles are spin $ \frac {1}{2}$ fermions that  satisfy the Pauli equation and are in strong contact we find infinitely many bound states on each cell of the  graph; when occupied they form the Fermi sea. At the surface these bound states are very  delocalized; an infinitesimal perturbation  gives an excited state with Dirac spectrum. 

We prove that in all dimensions both  strong and weak contact self-adjoint extensions are limit when $ \epsilon \to  0$ , in strong resolvent sense, of two body interactions through potentials $ V^\epsilon $ of decreasing support and of increasing strength (differently in the two cases).  

For the proofs we map the space into  one in which the functions are more singular;  the map can be interpreted as  fractioning and mixing. The new space has some relations with  semiclassical analysis. 
To come back to the original space we use Gamma convergence, a non perturbative variational method  of common use in the theory of  finely mixed composites.   


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