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Martin Ravn Christiansen

Publications and source records attributed to Martin Ravn Christiansen.

7 recordsLinked to original sources
arXiv2026

The Correlation Energy of the Electron Gas in the Mean-Field Regime

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 $c_{1}ρ\log\left(ρ\right)+c_{2}ρ$ 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.

math-ph↗
arXiv2025

A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

We prove that the eigenvalues of a 2-body operator $γ_{2}^Ψ$ associated to a fermionic $N$-particle state $Ψ$ are highly constrained by the structure of the corresponding eigenvectors: If $Φ=\sum_{k=1}^{\infty}λ_{k}u_{k}\wedge v_{k}$ is the canonical form of an eigenvector $Φ$ with eigenvalue $Λ$, then $Λ\leq(1+\frac{N-2}{2}\sum_{k=1}^{\infty}λ_{k}^{4})^{-1}N$. We also prove a lower bound on $\sup_{\Vert Ψ\Vert =1}\langle Φ,γ_{2}^ΨΦ\rangle$ for fixed $Φ$, and state a conjecture motivated by these results.

math-ph↗
arXiv2024

Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

We prove that the 2-body operator $γ_2^Ψ$ of a fermionic $N$-particle state $Ψ$ obeys $||γ_2^Ψ||_{HS} \leq \sqrt{5} N$, which complements the bound of Yang that $||γ_2^Ψ||_{op} \leq N$. 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 $γ_2^{Ψ,T}$ obeys the inequality $||γ_2^{Ψ,T}||_{HS} \leq \sqrt{5 N \, \mathrm{tr}(γ_1^Ψ(1 - γ_1^Ψ))}$.

math-ph↗
arXiv2023

The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime

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.

math-ph↗
arXiv2023

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

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 $c_{1}ρ\log\left(ρ\right)+c_{2}ρ$ 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.

math-ph↗
arXiv2023

Emergent Quasi-Bosonicity in Interacting Fermi Gases

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].

math-ph↗