[en] We discuss the ground state and the small-amplitude excitations of a degenerate vapour of fermionic atoms placed in two hyperfine states inside a spherical harmonic trap. An equations-of-motion approach is set up to discuss the hydrodynamic dissipation processes from the interactions between the two components of the fluid beyond mean-field theory and to emphasize analogies with spin dynamics and spin diffusion in a homogeneous Fermi liquid. The conditions for the establishment of a collisional regime via scattering against cold-atom impurities are analyzed. The equilibrium density profiles are then calculated for a two-component vapour of ⁴⁰K atoms: they are little modified by the interactions for presently relevant values of the system parameters, but spatial separation of the two components will spontaneously arise as the number of atoms in the trap is increased. The eigenmodes of collective oscillation in both the total particle number density and the concentration density are evaluated analytically in the special case of a symmetric two-component vapour in the collisional regime. The dispersion relation of the surface modes for the total particle density reduces in this case to that of a one-component Fermi vapour, whereas the frequencies of all other modes are shifted by the interactions. (author)

[en] We formulate integral boundary conditions for the time-dependent Schroedinger equation describing an atom with the laser interaction in the dipole approximation. The boundary conditions are imposed on the solution on a surface (boundary) which may be at a finite (but sufficiently remote) distance from the atom. For the numerical integration of the Schroedinger equation, these exact conditions may be used to replace diffuse absorbing potentials or mask functions. These later are usually introduced in order to (approximate) compensate unphysical reflection which occurs at the boundary of a finite region when a zero-value condition is imposed there on the solution. The method allows to substantially reduce the size of the space domain where integration is carried out numerically. Considering the numerical solution of the same one-dimensional model as that discussed by Boucke et al. (1998), we demonstrate the success of our approach

[en] We construct a semiclassical expression for the Husimi function of autonomous systems in one degree of freedom, by smoothing with a Gaussian function an expression that captures the essential features of the Wigner function in the semiclassical limit. Our approximation reveals the center and chord structure that the Husimi function inherits from the Wigner function, down to very shallow valleys, where lie the Husimi zeros. This explanation for the distribution of zeros along curves relies on the geometry of the classical torus, rather the complex analytic properties of the WKB method in the Bargmann representation. We evaluate the zeros for several examples. (author)

[en] Use of a new expansion for Coulomb spheroidal functions together with already known expansions for them in a more convenient variant and also introduction of a new diagonal operator allow one to calculate eigenfunctions integrals of the two-centre problem analytically, in a simpler way. Due to that it requires considerably less time for calculation to obtain various matrix elements of discrete spectrum of the two-centre problem which are used for the solution of physical problems. (author)

[en] The generalized technique of Bargmann potentials is elaborated for the reconstruction of time-dependent and time-independent two-dimensional potentials and the corresponding solutions in a closed analytic form on the basis of the inverse scattering problem in the adiabatic representation. A number of specific examples are considered within the parametric problem on the entire line and on the half-line. Matrix elements of an exchange interaction are calculated and studied in terms of obtained exact solutions of the parametric inverse problems

[en] The externally driven damped nonlinear Schroedinger (NLS) equation on the infinite line is studied. Existence and stability chart for its soliton solution is constructed on the plane of two control parameters, the forcing amplitude h and dissipation coefficient γ. For generic values of h and γ there are two coexisting solitons one of which (ψ₊) is always unstable. The bifurcation diagram of the second solution (ψ_-) depends on the dissipation coefficient: if γ<γ_cr, the ψ_- is stable for small h and loses its stability via a Hopf bifurcation as h is increased; if γ>γ_cr, the ψ_- is stable for all h. There are no 'stability windows' in the unstable region. We show that the previously reported 'stability windows' occur only when the equation is considered on a finite (and small) spatial interval

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[en] The aim of this paper is to summarize some recently obtained relations between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It has been shown that solutions of finite subsystems of the ALH can be used to derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear Schroedinger, Davey-Stewartson, Kadomtsev-Petviashvili (DP) and some other equations. Similar approach has been used to construct new integrable models: O(3,1) and multi field sigma models. Such 'universality' of the ALH becomes more transparent in the framework of the Hirota's bilinear method. The ALH, which is usually considered as an infinite set of differential-difference equations, has been presented as a finite system of functional-difference equations, which can be viewed as a generalization of the famous bilinear identities for the KP tau-functions. (author)

[en] The probability of observation of thermal and nuclear anomalies in PdD_x is calculated assuming a physical mechanism of phonon laser type to take place. It is shown that poor reproducibility of these results has natural quantum mechanical explanation. (author)