  We review our results on a mathematical dynamical theory for observables for
open many-body quantum nonlinear bosonic systems for a very general class of
Hamiltonians. We show that non-quadratic (nonlinear) terms in a Hamiltonian
provide a singular "quantum" perturbation for observables in some "mesoscopic"
region of parameters. In particular, quantum effects result in secular terms in
the dynamical evolution, that grow in time. We argue that even for open quantum
nonlinear systems in the deep quasi-classical region, these quantum effects can
survive after decoherence and relaxation processes take place. We demonstrate
that these quantum effects in open quantum systems can be observed, for
example, in the frequency Fourier spectrum of the dynamical observables, or in
the corresponding spectral density of noise. Estimates are presented for
Bose-Einstein condensates, low temperature mechanical resonators, and nonlinear
optical systems prepared in large amplitude coherent states. In particular, we
show that for Bose-Einstein condensate systems the characteristic time of
deviation of quantum dynamics for observables from the corresponding classical
dynamics coincides with the characteristic time-scale of the well-known quantum
nonlinear effect of phase diffusion.
