arXiv:2510.21268

Ground state energy of a dilute inhomogeneous Fermi gas

Thomas Gamet

Published Oct 24, 2025• 0 citations

Abstract

We study the ground state energy of a system of N fermions with two spin states in the large N limit. The particles are placed in an inhomogeneous trapping potential and interact via scaled interactions. We study a dilute limit where the range of the interaction potential is much smaller than the typical inter-particle distance. We show that the energy per particle converges to the Thomas-Fermi energy of the system, with a perturbative term corresponding tot he interaction and exhibiting the scattering length of the potential. The proof is decomposed into two bounds. First, we construct an appropriate test-state to prove the upper bound. Then, we prove the lower bound by the Dyson lemma, which allows us to regularize the interaction potential, and several semi-classical tools.

DOI: 10.48550/arxiv.2510.21268

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arXiv:2510.21268

Ground state energy of a dilute inhomogeneous Fermi gas

Thomas Gamet

Published Oct 24, 2025• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2510.21268

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p

-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].

Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin

1/2

fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.

N

fermions interacting with a short-range, positive potential of scattering length

a

. For spin 1/2 fermions, this is

E \sim E^0 + (\hbar^2/2m) 2 πN ρa

, where

E^0

is the energy of the non-interacting system and

ρ

is the density.

Abstract We prove an upper bound on the energy density of the dilute spin-

\frac{1}{2}

1 2 Fermi gas capturing the leading correction to the kinetic energy

8\pi a \rho _\uparrow \rho _\downarrow

8 π a ρ ↑ ρ ↓ with an error of size smaller than

a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }

a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any

\varepsilon > 0

ε > 0 , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3 ).