# Ground State Energy of Dense Gases of Strongly Interacting Fermions

## TL;DR

This paper calculates the ground state energy of strongly interacting fermions in a confined space using a modified bosonization technique.

## Contribution

The paper extends bosonization techniques to strongly interacting fermions in arbitrary dimensions.

## Key findings

- The ground state energy is analyzed for strongly interacting fermions with a specific interaction scaling.
- A transition in behavior is observed at the mean-field scaling threshold.
- Results contrast with known outcomes in the weakly interacting regime.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$N^{-\alpha }v$$\end{document}N-αv, where \documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in \mathbb {R}$$\end{document}α∈R and v is a function of positive type satisfying a mild regularity assumption. Our focus is on the strongly interacting case \documentclass[12pt]{minimal}
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\begin{document}$$\alpha <1-\frac{2}{d}$$\end{document}α<1-2d. We contrast our result with existing results in the weakly interacting case \documentclass[12pt]{minimal}
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\begin{document}$$\alpha >1-\frac{2}{d}$$\end{document}α>1-2d and the transition happening at the mean-field scaling \documentclass[12pt]{minimal}
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\begin{document}$$\alpha =1-\frac{2}{d}$$\end{document}α=1-2d. Our proof is an adaptation of the bosonization technique used to treat the mean-field case.