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