A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

Abstract

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

arXiv comment8 pages