Ground state of Bose gases interacting through singular potentials

We consider a system of $N$ bosons on the three-dimensional unit torus. The particles interact through repulsive pair interactions of the form $N^{3\beta -1}v(N^{\beta }x)$ for $\beta \in (0,1)$. We prove the next order correction to Bogoliubov theory for the ground state and the ground state energy.