A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

Abstract

We prove that the eigenvalues of a 2-body operator $γ_{2}^Ψ$ associated to a fermionic $N$-particle state $\hat{I}¨$ are highly constrained by the structure of the corresponding eigenvectors: If $\hat{I}¦={\sum }_{k=1}^{\infty }\hat{I}{»}_k{u}_k\wedge v_k$ is the canonical form of an eigenvector $\hat{I}¦$ with eigenvalue $\hat{I}›$, then $\hat{I}›\le (1+\frac{N-2}{2}{\sum }_{k=1}^{\infty }\hat{I}{»}_k^4{)}^{-1}N$. We also prove a lower bound on $\sup_{\Vert Ψ\Vert =1}\langle Φ,γ_{2}^ΨΦ\rangle$ for fixed $\hat{I}¦$, and state a conjecture motivated by these results.

8 pages