# The Gell-Mann$-$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

@inproceedings{Christiansen2022TheGF, title={The Gell-Mann\$-\$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime}, author={Martin Ravn Christiansen and Christian Hainzl and Phan Th{\`a}nh Nam}, year={2022} }

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρ log (ρ)+c2ρ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.

## One Citation

### On the effective quasi-bosonic Hamiltonian of the electron gas: collective excitations and plasmon modes

- PhysicsLetters in Mathematical Physics
- 2022

We consider an effective quasi-bosonic Hamiltonian of the electron gas which emerges naturally from the random phase approximation and describes the collective excitations of the gas. By a rigorous…

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