Dynamics of complex quantum systems with energy dissipation

TL;DR

This paper analyzes the statistical properties of eigenenergies and their second differences in complex quantum systems modeled by the Ginibre ensemble, revealing distribution laws and homogenization behavior.

Contribution

It provides analytical distributions for eigenenergy differences in Ginibre ensembles and formulates a homogenization law for eigenenergies.

Findings

Distributions of eigenenergy second differences are derived for N=3.

Analytical formulas for generic N-dimensional Ginibre ensemble are obtained.

Homogenization law of eigenenergies is established.

Abstract

A complex quantum system with energy dissipation is considered. The quantum Hamiltonians $H$ belong the complex Ginibre ensemble. The complex-valued eigenenergies $Z_{i}$ are random variables. The second differences $\Delta^{1} Z_{i}$ are also complex-valued random variables. The second differences have their real and imaginary parts and also radii (moduli) and main arguments (angles). For $N$=3 dimensional Ginibre ensemble the distributions of above random variables are provided whereas for generic $N$- dimensional Ginibre ensemble second difference distribution is analytically calculated. The law of homogenization of eigenergies is formulated. The analogy of Wigner and Dyson of Coulomb gas of electric charges is studied.