Second order energy expansion of Bose gases with three-body interactions

We provide a second order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud and Triay in 21, where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order N. In addition, we show that the ground state satisfies Bose-Einstein condensation with a rate of the order 1N.