Dynamics of complex quantum systems with energy dissipation

1 Abstract A complex quantum system with energy dissipation is considered. The quantum Hamilto-nians H belong the complex Ginibre ensemble. The complex-valued eigenenergies Z i are random variables. The second differences ∆ 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. 2 Introduction We study generic quantum statistical systems with energy dissipation. The quantum Hamil-tonian operator H is in given basis of Hilbert's space a matrix with random elements H ij [1, 2, 3]. The Hamiltonian H is not hermitean operator, thus its eigenenergies Z i are complex-valued random variables. We assume that distribution of H ij is governed by Ginibre ensemble [1, 2, 4, 5]. H belongs to general linear Lie group GL(N, C), where N is dimension and C is complex numbers field. Since H is not hermitean, therefore quantum system is dissipative system. Ginibre ensemble of random matrices is one of many Gaussian Random Matrix ensembles GRME. The above approach is an example of Random Matrix theory RMT [1, 2, 3]. The other RMT ensembles are for example Gaussian orthogonal ensemble GOE, unitary