Application of hermitean and nonhermitean random matrices to quantum statistical systems

TL;DR

This paper explores the properties of nonhermitean and hermitean random matrices in quantum systems, focusing on eigenenergy distributions, differences, and analogies with electrostatic models, extending the understanding of dissipative quantum dynamics.

Contribution

It introduces an analysis of complex eigenenergies in nonhermitean matrices and compares them with hermitean ensembles, extending the space of dynamics with discrete labeling indices.

Findings

Eigenenergies of nonhermitean matrices are complex-valued and exhibit specific statistical properties.

The second difference of eigenenergies acts as a discrete Hessian, revealing new insights into spectral fluctuations.

Comparison with Gaussian hermitean ensembles highlights differences in spectral behavior and dynamics.

Abstract

The Ginibre ensemble of nonhermitean random Hamiltonian matrices $K$ is considered. Each quantum system described by $K$ is a dissipative system and the eigenenergies $Z_{i}$ of the Hamiltonian are complex-valued random variables. The second difference of complex eigenenergies is viewed as discrete analog of Hessian with respect to labelling index. The results are considered in view of Wigner and Dyson's electrostatic analogy. An extension of space of dynamics of random magnitudes is performed by introduction of discrete space of labeling indices. The comparison with the Gaussian ensembles of random hermitean Hamiltonian matrices $H$ is performed.