Description of Quantum Systems by Random Matrix Ensembles of High Dimensions: ICSSUR'6 Poster Session

TL;DR

This paper introduces a theorem describing the maximum of the probability density functions of the second difference of energy levels in high-dimensional random matrix ensembles, revealing insights into level distribution behaviors.

Contribution

It formulates a new theorem on the location of the maximum of the second difference probability density functions for various high-dimensional ensembles, including GOE, GUE, GSE, and Poisson.

Findings

Maximum at the origin for ensembles with N ≥ 3

Introduction of level homogenization concepts

Theorem applicable to multiple ensembles

Abstract

The new Theorem on location of maximum of probability density functions of dimensionless second difference of the three adjacent energy levels for $N$-dimensional Gaussian orthogonal ensemble GOE($N$), $N$-dimensional Gaussian unitary ensemble GUE($N$), $N$-dimensional Gaussian symplectic ensemble GSE($N$), and Poisson ensemble PE, is formulated: {\it The probability density functions of the dimensionless second difference of the three adjacent energy levels take on maximum at the origin for the following ensembles: GOE($N$), GUE($N$), GSE($N$), and PE, where $N \geq 3$.} The notions of {\it level homogenization with level clustering} and {\it level homogenization with level repulsion} are introduced. [poster session of ICSSUR'6].