Vestnik of Samara University. Natural Science Series

The article is dedicated to the memory of Doctor of Physical and Mathematical Sciences, Professor Vladimir Ivanovich Astafiev, whose professional activity for more than 35 years has been associated with Samara University. Scientific, teaching and organizational activities of V.I. Astafiev largely determined and will determine the scientific directions developed at the Faculty of Mechanics and Mathematics, and the educational activity at the Faculty of Mechanics and Mathematics. His boundless dedication to the university, his broad and deep education, and high mathematical culture allowed him to educate the whole pleiades of scientists and professors currently working at the university.

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω.

Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.

An analytical method of solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximate
method was made depending on the speed of the boundary movement.

A set of modeling programs for constructing a sequence of energy zones of heterojunctions is presented for analyzing the distribution of charge carriers in the heterostructure and internal characteristics, for describing the processes of charge transfer and accumulation. Wolfram Mathematica analytical system and Delphi programming language were used. The main elements of materials are semiconductors, metals of contact structures and injection regions of nonequilibrium carriers. The programs allow determining the structural characteristics of materials, active zones and spatial charge regions, calculating quasi-Fermi levels and built-in potentials, as well as the efficiency of heterostructures in general and for separation-charge collection, charge accumulation, determining the type of metallization of barrier or ohmic contact.

In this paper, we investigated the dynamics of systems of two and three identical qubits interacting resonantly with a selected mode of a thermal field of a lossless resonator. We found solutions of the quantum time-dependent Liouville equation for various three- and two-qubit entangled states of qubits. Based on these solutions, we calculated the criterion of the qubit entanglement — fidelity. The results of numerical calculations of the fidelity showed that increasing the average number of photons in a mode leads to a decrease in the maximum degree of entanglement. It is shown that the two-qubit entangled state is more stable with respect to external noise than the three-qubit entangled Greenberger — Horne — Zeilinger states (GHZ). Moreover, a genuine entangled GHZ-state is more stable to noise than a GHZ-like entangled state.

The exact dynamics of a model consisting of two two-level atoms interacting with the electromagnetic field mode of an ideal resonator through degenerate Raman transitions are found for coherent and thermal field states. The exact solution is used to calculate atom-atom negativity. It is shown that for separable initial states of atoms, their interaction with the resonator field does not lead to the occurrence of atom-atom entanglement. It was found that for the Bell initial states of atoms in the case of a coherent resonator field, the effect of sudden death of entanglement takes place for large average values of the number of photons in the resonator, while in case of thermal noise, this effect is absent for any intensities of the resonator field.