Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

Abstract

We prove that the 2-body operator ${\gamma }_2^{\Psi }$ of a fermionic $N$-particle state $\Psi$ obeys $||{\gamma }_2^{\Psi }|{|}_{HS}\le \sqrt{5}N$, which complements the bound of Yang that $||{\gamma }_2^{\Psi }|{|}_{op}\le N$. 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 $||{\gamma }_2^{\Psi ,T}|{|}_{HS}\le \sqrt{5N\mathrm{tr}({\gamma }_1^{\Psi }(1-{\gamma }_1^{\Psi }))}$.

8 pages, final version

References in corpus (1)

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