Abstract:
High approximations of semiclassical trajectory-coherent states (TCS) and of semiclassical Green function (in the class of semiclassically concentrated states) for the Dirac operator with anomalous Pauli interaction are obtained. For Schrodinger and Dirac operators trajectory-coherent representations are constructed up to any precision with respect to h, h-->0.

Abstract:
Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate as $\h\to 0$ a quantum-mechanical state. This idea leads to a family of systems of ordinary differential equations, called Ehrenfest M-systems (M=0,1,2,...), formally equivalent to the semiclassical approximation for the linear Schroedinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Ehrenfest systems, without solving the quantum equation: the semiclassical asymptotics for the spectrum are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.

Abstract:
We study a Hartree ensemble approximation for real-time dynamics in the toy model of 1+1 dimensional scalar field theory. Damping behavior seen in numerical simulations is compared with analytical predictions based on perturbation theory in the original (non-Hartree-approximated) model.

Abstract:
In this paper we construct the coherent and trajectory-coherent states of a damped harmonic oscillator. We investigate the properties of this states.

Abstract:
We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets, and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.

Abstract:
In this paper we construct the trajectory-coherent states for the Caldirola - Kanai Hamiltonian. We investigate the properties of this states.

Abstract:
In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion.

Abstract:
The coherent potential approximation (CPA) is extended to describe satisfactorily the motion of particles in a random potential which is spatially correlated and smoothly varying. In contrast to existing cluster-CPA methods, the present scheme preserves the simplicity of the conventional CPA in using a single self-energy function. Its accuracy is checked by a comparison with the exact moments of the Green's function, and with the spectral function from numerical simulations. The scheme is applied to excitonic absorption spectra in different spatial dimensions.

Abstract:
In this paper we investigate the problem of minimization the Heisenberg's uncertainty relation by the trajectory-coherent states. The conditions of minimization for Hamiltonian and trajectory are obtained. We show that the trajectory-coherent states minimize the Heisenberg's uncertainty relation for special Cauchy problem for the Schr\"{o}dinger equation only.

Abstract:
We study the dynamics of a spatially inhomogeneous quantum $\lambda \phi^4$ field theory in 1+1 dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case. Nevertheless, we show that the fields fail to thermalize in a canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasi-classical inhomogeneous configurations (e.g. solitons) in a background of dynamical self-consistent quantum fluctuations.