Search

We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained in \cite{ChrHaiNam-23b}, our result establishes an analogue of the Gell-Mann--Brueckner formula $c_{1}\rho\log\left(\rho\right)+c_{2}\rho$ for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials., Comment: 60 pages

We prove that the 2-body operator $\gamma_2^\Psi$ of a fermionic $N$-particle state $\Psi$ obeys $||\gamma_2^\Psi||_{HS} \leq \sqrt{5} N$, which complements the bound of Yang that $||\gamma_2^\Psi||_{op} \leq N$. This estimate furthermore resolves a conjecture of Carlen-Lieb-Reuvers (arXiv:1403.3816) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert-Schmidt norm of the truncated 2-body operator $\gamma_2^{\Psi,T}$ obeys the inequality $||\gamma_2^{\Psi,T}||_{HS} \leq \sqrt{5 N \, \mathrm{tr}(\gamma_1^\Psi (1 - \gamma_1^\Psi))}$., Comment: 8 pages, final version

This thesis concerns the correlation structure of interacting Fermi gases on a torus in the mean-field regime. A bosonization method in the spirit of Sawada is developed to analyze the system, and is applied to obtain an upper bound for the correlation energy of the system for a wide class of repulsive interaction potentials, including the Coulomb potential. This upper bound includes both a bosonic contribution, as found in the bosonic model of Sawada, and an exchange contribution, as was found by Gell-Mann and Brueckner but which was missed by Sawada's model. An extension to weakly attractive potentials is also presented, as is an outline of the derivation of an effective Hamiltonian for regular interaction potentials, and the construction of plasmon states for this outside of the mean-field setting. This thesis is based on the papers [arXiv:2106.11161, arXiv:2206.13073, arXiv:2208.01581]., Comment: PhD thesis, 141 pages

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 $c_{1}\rho\log\left(\rho\right)+c_{2}\rho$ 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., Comment: 50 pages, final version

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 argument, we explain how the plasmon modes can be interpreted as a special class of approximate eigenstates of this model., Comment: 34 pages, 1 figure, final version

We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of $N$ fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-\frac{1}{3}}$. In the limit $N\to\infty$, we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature., Comment: Final version to appear in Forum Math. Pi (124 pages), plus a supplementary note (15 pages)