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. 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.

A new method for diagonalizing the nuclear shell model Hamiltonians with large dimensions

“…Indeed the correlation is remarkably good. We found that this correlation is well applicable not only to realistic systems, but also to any two-body random ensemble [3]. Below we give a few examples.…”

“…An example is shown on the left side of Fig. 1, which shows the distribution of eigenvalues E First we sort the matrix elements H (I) i j in such a way that the diagonal elements are sorted from smaller values to larger ones [3]:…”

Regularity of nuclear structure under random interactions

“…Mr. Shen started this practice when he was an undergraduate student in Shanghai. He made two interesting progresses in his undergraduate thesis [14]: (1) he considered the third moment and improved our statistical formula. (2) He accidentally noticed an interesting correlation between eigenvalues and diagonal matrix elements for a shell model Hamiltonian, after sorting the diagonal matrix elements from the smaller to the larger.…”

How random are matrix elements of the nuclear shell model Hamiltonian?

“…[10,11], we sorted diagonal matrix elements from the smaller to larger ones, and found a very strong (statistical) linear correlation between exact eigenvalues E i and diagonal matrix elements H…”

“…Along the same line, Yoshinaga et al [6,7] studied the lowest energy of spin I states by the moments of energy spectra based on the fact that the eigenvalues of Hamiltonian under two-body random ensemble follow the Gaussian distribution [8,9] . Recently, we found linear (statistical) correlation between eigenvalues and diagonal matrix elements in the nuclear shell model Hamiltonian [10,11] .…”