An example of dissipative quantum system: finite differences for complex Ginibre ensemble

TL;DR

This paper analyzes the eigenenergy differences in the complex Ginibre ensemble, providing analytical formulas for their distributions and exploring their implications for dissipative quantum systems.

Contribution

It introduces analytical formulas for the distribution of eigenenergy second differences in the Ginibre ensemble, advancing understanding of dissipative quantum systems.

Findings

Distribution formulas for eigenenergy differences derived

Real and imaginary parts analyzed for N=3

Homogenization law of eigenenergies formulated

Abstract

The Ginibre ensemble of complex random Hamiltonian matrices $H$ is considered. Each quantum system described by $H$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. For generic $N$-dimensional Ginibre ensemble analytical formula for distribution of second difference $\Delta^{1} Z_{i}$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $\Delta^{1} Z_{i}$ and also of its modulus and phase are provided for $N$=3. The results are considered in view of Wigner and Dyson's electrostatic analogy. General law of homogenization of eigenergies for different random matrix ensembles is formulated.