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

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.

arXiv comment5 pages; presented at poster session of the conference "Quantum dynamics in terms of phase-space distributions"; May 22nd, 2000 to May 26th, 2000; Max Planck Institute for the Physics of Complex Systems; Dresden, Germany (2000)