A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

We prove that the eigenvalues of a 2-body operator ${\gamma }_2^{\mathrm{\Psi }}$ associated to a fermionic $N$-particle state $\mathrm{\Psi }$ are highly constrained by the structure of the corresponding eigenvectors: If $\mathrm{\Phi }={\sum }_{k=1}^{\mathrm{\infty }}{\lambda }_k{u}_k\wedge v_k$ is the canonical form of an eigenvector $\mathrm{\Phi }$ with eigenvalue $\mathrm{\Lambda }$, then $\mathrm{\Lambda }\le (1+\frac{N-2}{2}{\sum }_{k=1}^{\mathrm{\infty }}{\lambda }_k^4{)}^{-1}N$. We also prove a lower bound on ${\sup }_{\mathrm{\parallel }\mathrm{\Psi }\mathrm{\parallel }=1}⟨\mathrm{\Phi },{\gamma }_2^{\mathrm{\Psi }}\mathrm{\Phi }⟩$ for fixed $\mathrm{\Phi }$, and state a conjecture motivated by these results.