Zero range (contact) interactions conspire to produce Efimov trimers and quadrimers

We introduce \emph{contact (zero range) interactions } , a special class of self-adjoint extensions of the N-body Schr\"odinger free hamiltonian $H_0$ restricted to functions with support away from the \emph{contact manifold} $\Gamma \equiv \cup \Gamma_{i,j} \;\;\; \Gamma_{i,j}\equiv \{x_i = x_j \; i \not= j \} \;,\; x_i \in R^3$. These extensions are defined by boundary conditions at $\Gamma$. We discuss the spectral properties as function of the masses and the statistics The (Efimov) spectrum is entirely due "conspiracy" of the contact interactions of two pairs. These states are called in Theoretical Physics \emph{trimers} if the two pairs have an element in common, \emph{quadrimers} otherwise. The analysis can be extended to the case in which there is a regular two-body potential, but then the spectral properties cannot be given explicitly. We prove that these interactions are limit (in strong resolvent sense) when $\epsilon \to 0$ of N-body hamiltonians $H_\epsilon = H_0+ \sum_{i,j} \frac {1}{\epsilon^3}V_{i,j}( \frac {|x_i-x_j|}{\epsilon}) ,\; V_{i,j} \in L^1$. The result is \emph{ independent of the shape of the potential} Strong resolvent convergence implies convergence of spectra.This makes contact interaction a valuable tool in the study of the spectrum of a system of $N$ particles interacting through potentials of very short range.