Motivated by the Jaynes-Cummings (JC) model, we consider here a quantum dot coupled simultaneously to a reservoir of photons and to two electric leads (free-fermion reservoirs). This new Jaynes-Cummings-Leads (JCL) model permits a fermion current through the dot to create a photon flux, which describes a light-emitting device. The same model is also used to describe the transformation of a photon flux into a current of fermions, i.e. a quantum dot light-absorbing device. The key tool to obtain these results is the abstract Landauer-Büttiker formula.

Let R² be the exterior of a convex polygon whose side lengths are `1; : : : ; `M. For a real constant  , let H   denote the Laplacian in , u 7! 􀀀u, with the Robin boundary conditions @u=@ =  u at @ , where is the outer unit normal. We show that, for any fixed m 2 N, the mth eigenvalue E m ( ) of H   behaves as E m ( ) = 􀀀 2 + D m + O( 􀀀1=2) as   ! +1, where D m stands for the mth eigenvalue of the operator D1 DM and Dn denotes the onedimensional Laplacian f 7! 􀀀f00 on (0; `n) with the Dirichlet boundary conditions.

Thermally induced magnetic transitions are rare events as compared with vibrations of individual magnetic moments. Timescales for these processes differ by 10 orders of magnitude or more. Therefore, the standard MonteCarlo simulation is not suitable for the theoretical description of such phenomena. However, a statistical approach based on transition state theory is applicable for calculations of the transition rates. It presupposes finding the minimum energy path (MEP) between stable magnetic states on the multidimensional energy surface of the system. A modification of the Nudged Elastic Band (NEB) method for finding the energy barriers between states is suggested. A barrier on the energy surface corresponds to the difference between maximum energy along the MEP (highest saddle point) and the initial state minimum. The NEB procedure is implemented for spin rotations in Cartesian representation with geometric constraint on the magnitude of the magnetic moment. In this case, the effective magnetic forces are restricted to the tangent plane of the magnetic momentum vector.

We consider applications of the Boundary Control (BC) method to generalized spectral estimation problems and to inverse source problems. We derive the equations of the BC method for these problems and show that the solvability of these equations crucially depends on the controllability properties of the corresponding dynamical system and properties of the corresponding families of exponentials.

In this report we discuss the problem of approximating nonlinear delta-interactions in dimensions one and three with regular, local or non-local nonlinearities. Concerning the one dimensional case, we discuss a recent result proved in [10], on the derivation of nonlinear delta-interactions as limit of scaled, local nonlinearities. For the three dimensional case, we consider an equation with scaled, non-local nonlinearity. We conjecture that such an equation approximates the nonlinear delta-interaction, and give an heuristic argument to support our conjecture.

Quantum random number generation allows the obtaining of true random numbers that can be used for applications (e.g.,a cryptography) requiring a high degree of randomness. In this paper, we give a mathematical description of a quantum random number generation system using homodyne detection. As a result of the theoretical research, we obtained the description of the relationship between beam splitter input radiation and di erential current on detectors after beam splitting. We derived equations allowing one to estimate the scheme parameters imperfection impact on measurement results. We also obtained mathematical expres- sions, demonstrating the equivalence of quantum description of Y-splitter and beam splitter with two inputs, which allows the use Y-splitter for the implementation of quantum random number generation systems based on vacuum quantum 
uctuations.

The e ects of the local accumulation of charges in resonant tunnelling heterostructures have been described using 1D Shrodinger-Poisson Hamiltonians in the asymptotic regime of quantum wells. Taking into account the features of the underling physical system, the corresponding linearized model is naturally related to the adiabatic evolution of shape resonances on a time scale which is exponentially large w.r.t. the asymptotic parameter h. A possible strategy to investigate this problem consists of using a complex dilation to identify the resonances with the eigenvalues of a deformed operator. Then, the adiabatic evolution problem for a sheet-density of charges can be reformulated using the deformed dynamical system which, under suitable initial conditions, is expected to evolve following the instantaneous resonant states. 
After recalling the main technical diculties related to this approach, we introduce a modi ed model where h-dependent arti cial interface conditions, occurring at the boundary of the interaction region, allow one to obtain adiabatic approximations for the relevant resonant states, while producing a small perturbation of the dynamics on the scale hN0 . According to these results, we  nally suggest an alternative formulation of the adiabatic problem. An a posteriori justi cation of our method is obtained by considering an explicitlysolvable case.

A full asymptotic series for low eigenvalues and eigenfunctions of a stationary Schr¨odinger operator with a nondegenerate well was constructed in [29]. This allowed us to describe the tunneling effect for a potential with two or more identical wells with sufficient accuracy. The procedure is described in the following discussion. Some formulae are obtained and corresponding problems are discussed.

Numerical solutions of the Liouville equation are used to study of the polarization characteristics of an ensemble of He atoms in 3³S₁ state in the presence of the strong magnetic field when the coherent population of this level is realized from 2³P state.

A brief review of effects in information recording systems based on complex compounds of polyvinyl alcohol (PVA) with metals (Au, Cu, Pt, Bi) is presented. As the result of irradiation, the chain reaction process of decomposition causes the aggregation of metal atoms or metal salts molecules to point centers or dendritic crystals. Some features of these processes are outlined.

The Stokes approximation is used for the description of flow in nanostructures. An algorithm for Stokes  ow computation in cases when there is great variation in the viscosity over a small spatial region is described. This method allows us to overcome computational dificulties of the  nite-dierence method. The background of the approach is using the Woodbury formula - a discrete analog of the Krein resolvent formula. The particular example of a rectangular domain is considered in detail. The inversion of the discrete Stokes operator is made in analytic form for the case of constant viscosity.

This paper is devoted to strong solutions of the first and second initialboundary problems for nonuniformly parabolic equations. These equations are used in mechanics, glaciology, rheology, image processing as well as for nanosystem modeling. The initial data space for these problems was explicitly described as Orlicz—Sobolev spaces.