Large particle number asymptotics for point-interacting bosons in the plane

Abstract

We consider $N$ bosons in two dimensions interacting via point interactions, modeled by the TMS Hamiltonian. We prove that their ground state energy has the asymptotic behavior $-e^{C N }$ for large $N$, for an explicit constant $C$ related to the sharp constant in the Ladyzhenskaya-Gagliardo-Nirenberg inequality. Such asymptotic behavior was predicted in the physics literature, based on a variant of the mean-field approximation.