Quantum symmetrical statistical system: Ginibre-Girko ensemble

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 ${\mathrm{\Delta }}^1{Z}_i$ of complex eigenenergies is presented. The distributions of real and imaginary parts of $\Delta^{1} Z_{i}$andalsoofitsmodulusandphaseareprovidedfor$N$=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.