Strong Linear Correlation Between Eigenvalues and Diagonal Matrix Elements

TL;DR

This paper reveals a strong linear correlation between eigenvalues and diagonal matrix elements in many-body systems and random matrices, enabling efficient eigenvalue prediction without complex computations.

Contribution

It uncovers a novel linear correlation that allows for simplified eigenvalue estimation in complex quantum systems and random matrices.

Findings

Strong linear correlation between eigenvalues and diagonal elements.

Eigenvalues can be predicted accurately using this correlation.

Method reduces computational complexity for eigenvalue estimation.

Abstract

We investigate eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices. We find a strong linear correlation between eigenvalues and diagonal matrix elements if both of them are sorted from the smaller values to larger ones. By using this linear correlation we are able to predict reasonably all eigenvalues of given shell model Hamiltonian without complicated iterations.