Ground state energy of dense gases of strongly interacting fermions

We study the ground state energy of a gas of $N$ fermions confined to a unit box in $d$ dimensions. The particles interact through a 2-body potential with strength scaled in an $N$-dependent way as $N^{-\alpha }v$, where $\alpha \in \mathbb{R}$ and $v$ is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case $\alpha <1-\frac{2}{d}$. We contrast our result with existing results in the weakly interacting case $\alpha >1-\frac{2}{d}$, and the transition happening at the mean-field scaling $\alpha =1-\frac{2}{d}$. Our proof is an adaptation of the bosonization technique used to treat the mean-field case.