ON THE AVERAGED QUANTUM DYNAMICS BY WHITE-NOISE HAMILTONIANS WITH ANDWITHOUT DISSIPATION

Exact results are derived on the averaged dynamics of a class of rando m quantum-dynamical systems in continuous space. Each member of the cl ass is characterized by a Hamiltonian which is the sum of two parts. W hile one part is deterministic, time-independent and quadratic, the We yl-Wigner symbol of the other part is a homogeneous Gaussian random fi eld which is delta correlated in time, but smoothly correlated in posi tion and momentum. The averaged dynamics of the resulting white-noise system is shown to be a monotone mixing increasing quantum-dynamical s emigroup. Its generator is computed explicitly. Typically, in the cour se of time the mean energy of such a system grows linearly to infinity . In the: second part of the paper an extended model is studied, which , in addition, accounts for dissipation by coupling the white-noise sy stem linearly to a quantum-mechanical harmonic heat bath. It is demons trated that, under suitable assumptions on the spectral density of the heat bath. the mean energy then saturates for long times.