A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

TL;DR

This paper establishes bounds on the eigenvalues of fermionic 2-body operators based on the structure of their eigenvectors, providing new theoretical constraints relevant to quantum many-body physics.

Contribution

It introduces a novel correlational bound for eigenvalues of fermionic 2-body operators, linking eigenvalues to eigenvector structure and proposing a related conjecture.

Findings

Eigenvalues are constrained by eigenvector structure.

Derived an upper bound involving eigenvector coefficients.

Proposed a lower bound and a conjecture related to these bounds.

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 $\Lambda\leq(1+\frac{N-2}{2}\sum_{k=1}^{\infty}\lambda_{k}^{4})^{-1}N$. We also prove a lower bound on $\sup_{\Vert \Psi\Vert =1}\langle \Phi,\gamma_{2}^{\Psi}\Phi\rangle$ for fixed $\Phi$, and state a conjecture motivated by these results.