Contact Interactions and Gamma Convergence - Dell'Antonio, Gianfausto (Frontiers in Physics, 2019)
We introduce contact interactions defined by boundary conditions at the contact manifold Γ≡∪i, j{xi = xj}. There are two types of contact interactions, weak and strong. Both provide self-adjoint extensions of Ĥ0 the free hamiltonian restricted away from Γ. We analyze both of them by “lifting” the system to a space of more singular functions: the map is fractioning and mixing. In the new space we use tools of Functional Analysis. After returning to physical space we use Gamma convergence, a well-known variational tool. We prove that contact interactions are strong resolvent limits of potentials with finite range. Weak contact of one boson with two other bosons leads to the low-density Bose-Einstrin condensate. Simultaneous weak contact of three bosons produces the high-density condensate which has an Efimov sequence of bound states. In Low Energy Physics strong contact of one particle with another two produces an Efimov sequence of bound states (we will comment briefly on the relation with the effect with the same name in Quantum Mechanics). For N bosons strong contact gives a lower bound −CN for the energy. A system of fermions in strong contact (unitary gas) has a positive hamiltonian. We give several examples in dimension 3,2,1. In the Appendix we describe the ground state of the Polaron.