Galerkin’s Method in the Theory of Dipole Antennas

It is proved that an approximate solution of the integro-differential equation for the current at the surface of a dipole antenna obtained by Galerkin’s method converges to the exact solution. A method is proposed for calculating the matrix elements of linear operators that enter the above equation.     

Fluctuations of the thermodynamic properties of a magnetized quantum cylinder in the vicinity of the critical temperature

The analytical formulas that describe the dependence of the fluctuation contribution to the thermodynamic properties of a magnetized quantum cylinder on its characteristic sizes are given. It is shown that the role of fluctuation cylinders substantially increases as the system dimensionality increases.     

Spin and dispersion of a massive Dirac neutrino in a magnetized plasma

The total energy shift of a polarized massive Dirac neutrino in an electron–positron plasma in a constant magnetic field is investigated. The calculation in the Feynman gauge is performed for the first time by using the Matsubara temperature Green functions. The dependence of the dispersion relation and the anomalous magnetic moment of the neutrino on the magnetic field strength, spin, energy, direction of motion, neutrino mass, and the plasma parameters is analyzed. The results of investigations for the massive neutrino in the limiting case are compared with those obtained earlier by other authors for the massless left neutrino.     

Theory of Integral Equations for Axisymmetric Scattering by a Disk

Computational Mathematics and Mathematical Physics - A theory of integral equations for radial currents in the axisymmetric problem of scattering by a disk is constructed. The theory relies on the...     

Superconductivity of the magnetized electron gas of a quantum cylinder

A microscopic theory of superconductivity is developed for the magnetized electron gas on a cylindrical surface. The Gibbs free energy is calculated for the superconducting system. A gap equation is derived that determines the critical temperature as a function of the quantum-cylinder dimensions and the Aharonov-Bohm parameter. It is shown that the gap not only exhibits Aharonov—Bohm oscillations, but also oscillates with varying curvature of the cylindrical surface.     

Template-Free Hydrothermal Synthesis of Hollow α-FeOOH Urchin-Like Spheres and Their Conversion to α-Fe2O3 Under Low-Temperature Thermal Treatment in Air

Hollow α-FeOOH urchin-like spheres were synthesized by a simple hydrothermal method at 160 °C for 12 h and their thermal conversion to hollow α-Fe₂O₃ urchin-like spheres was performed at 300 °C for 2 h in air. The results from X-ray diffraction and electron microscopy analyses reveal that hollow α-FeOOH urchin-like spheres were completely transformed to hollow α-Fe₂O₃ urchin-like spheres without a significant morphological change. Also, the effect of hydrothermal treatment temperature (170–200 °C for 12 h) on the phase structure and morphology of the final product was investigated. Pure α-FeOOH, the mixture of α-FeOOH and α-Fe₂O₃, and pure α-Fe₂O₃ with different morphologies were obtained at <180, 180–190 and 200 °C, respectively. The obtained materials can be used in the photodegradation of organic pollutants under visible light irradiation.     

Ionization of a two-dimensional quantum dot by the field of an electromagnetic wave

The process of ionization of a two-dimensional quantum dot by the field of a linearly polarized electromagnetic wave is studied. For the first time, analytic expressions for the ionization rate and partial probabilities of the process per unit time are obtained. The dependence of the probability of the process on the parameters of the confining potential and the Keldysh parameter are studied. The results of the work are compared with the earlier results on one- and three-dimensional nanostructures with a short-range potential.     

Justification of the Galerkin method for hypersingular equations

The paper presents a theoretical study of hypersingular equations of the general form for problems of electromagnetic-wave diffraction on open surfaces of revolution. Justification of the Galerkin is given. The method is based on the separation of the principal term and its analytic inversion. The inverse of the principal operator is completely continuous. On the basis of this result, the equivalence of the initial equation to a Fredholm integral equation of the second kind is proven. An example of numerical solution with the use of Chebyshev polynomials of the second kind is considered.