The condensation of a trapped dilute Bose gas with three-body interactions

TL;DR

This paper proves that in the large particle limit, a trapped dilute Bose gas with three-body interactions exhibits Bose--Einstein condensation and its ground states are described by a nonlinear Schrödinger functional.

Contribution

It establishes the connection between the many-body ground state and a nonlinear Schrödinger functional for three-body interactions in the dilute Bose gas.

Findings

Ground states are convex superpositions of NLS minimizers.

Complete Bose--Einstein condensation occurs under uniqueness of the NLS minimizer.

The nonlinear coupling constant relates to the scattering energy of the interaction potential.

Abstract

We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schr\"odinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the $N$-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schr\"odinger minimizer is unique up to a complex phase.