Abstract

We consider a trapped dilute gas of $N$ bosons in ${ℝ}^3$ interacting via a three-body interaction potential of the form $NV(N^{1∕2}(x-y,x-z))$. 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 Schrödinger 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 Schrödinger minimizer is unique up to a complex phase.

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