Strong correlation between eigenvalues and diagonal matrix elements

Abstract

We investigate eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices. For two-body random ensemble, 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 a given Hamiltonian matrix without complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangent function improves the accuracy of predicted eigenvalues near the minimum and maximum.