Contact Interactions and Gamma Convergence

Abstract

We study contact interactions, a generalization of Albeverio’s point interactions. There are two types of contact interactions, weak and strong; the last type occurs only in a three particle system. Strong contact leads to systems that have an infinite number of bound states with eigenvalues that decrease with a scaling law. We prove that in both the strong and the weak contact cases the hamiltonians are strong resolvent limits of hamiltonians with potentials with support that vanishes with a given scaling law while the $$L^1$$ norm remains constant. In the weak contact case, the approximating hamiltonians must have a zero energy resonance. As applications we describe Bose-Einstein condensation in the low and high density regimes, the Fermi sea in solid state physics and the ground state of Nelson’s polaron.