Contact interactions in Q.M. Gamma convergence and Bose-Einstein condensation

Abstract

We describe \emph{zero range} (contact) interactions, weak and strong. Strong contact leads to the Efimov effect in Low Energy Nuclear Physics, weak contact leads to Bose-Einstein condensation and to the presence of Cooper pairs in superconductivity . In the B-E. case we derive the Gross-Pitayewskii equation. For a condensate of Cooper pairs we derive a \emph{different} cubic local focusing P.D.E. We prove that both strong and weak contact hamiltonians are limits, in strong resolvent sense, of hamiltonians with potentials of very short range. A part form standard tools in Q.M. we use Gamma convergence Contact interactions and Gamma convergence have a role also in Solid State Physics for particles (conduction electrons) which satisfy the Pauli equation and move in the neighborhood of a lattice with $Y$-shaped vertices. The resulting \emph{Fermi sea} is the filling of an Efimov sequence of bound states with $ \frac{1}{\log n}$ rate of decrease of the negative eigenvalues. Contact interactions describe also a \emph{contact} (Nelson) Polaron (massive particle in (contact) interaction with a mass zero field), both relativistic and non-relativistic. We shall describe elsewhere these structures.