  We consider Bose gases of $N$ particles in a box of volume one, interacting
through a repulsive potential with scattering length of order $N^{1-\kappa}$,
for $\kappa > 0$. Such regimes interpolate between the Gross-Pitaevskii and
thermodynamic limits. Assuming that $\kappa$ is sufficiently small, we
determine the ground state energy and the low-energy excitation spectrum of the
system, up to errors vanishing in the limit of large $N$.
