The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Resonances are defined as the poles of the scattering matrix. The poles are associated with the complex eigenvalues of the Hamiltonian which are embedded in the lower half of the complex plane. The asymptotes of the corresponding eigenfunctions are exponentially diverged. Therefore, the resonance eigenfunctions are not embedded in the Hermitian domain of the Hamiltonian. The time asymmetric problem is discussed for these types of non-Hermitian Hamiltonians and several solutions of this problem are proposed.     

On the stability of some analytic results arising in the characterization of distributions

Zinger characterization describes a class of probability distributions that have finite moments of all orders. The stability problem is considered by the properties of independence of statistics in this characterization. It was proved by method of Azlarov that order statistics satisfying a certain ε-regression condition also have finite moments of all orders. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

The investigation of the photonic waveguide structure made of silicon carbide by means of the method of the singular integral equations

Here we present the analyses of the open (without a metal screen) ridged and photonic waveguide structures by means of the electrodynamical rigorous method of the Singular Integral Equations (SIE). The waveguides are made of a lossy silicon carbide (SiC) material. We have discovered peculiarities of the dispersion characteristics. We have found the numerical solutions to the complex wave equations. The dispersion characteristics of both waveguide structures are numerically analyzed and compared with each other. It was found that the losses of modes propagating in the waveguide structure with air openings (channels) are smaller than the losses in the waveguide structure without air openings. We came to the conclusion that it is possible to optimize the dispersion characteristics by adding openings of different shapes and sizes into the waveguide structures.     

Scattering characteristics of structures of lossy metamaterial–semiconductor cylinders

Here, we present for the first time the rigorous boundary problem solution of the Maxwell’s equations for the determination of scattering characteristics of a structure. The structure consists of a finite set of infinite parallel circular cylinders that can be made of different lossy isotropic materials. We numerically analyzed two structures that differ only in the symmetrical arrangement of semiconductor cylinders in relation to a central metamaterial cylinder. The electrical radii of cylinders can be arbitrary. Both polarizations of the incident microwave are considered in this work. The Poynting vector of the plane microwave that reflected from and transmitted through the structures analyzed here. We investigated dependency on the radius of an arc where are placed the semiconductor cylinders, the semiconductor-specific resistivity, the operating frequency at two radii of the metamaterial cylinder. We discovered that the structure can have features of a band gap photonic crystal dependent on the topology and the polarization of the incident microwave. We have found that the structure can operate as a microwave reflector at the certain radius of the arc on which are located thirteen n-Si cylinders. The Poynting vector is very sensitive to the change of semiconductor-specific resistivity when the incident microwave has the parallel polarization.     

On the stability of characterization of the normal distribution by the properties of linear forms

If the assumptions of the characterization theorem are fulfilled not exactly but only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems in which this kind of problems are considered are called stability theorems or the stability of characterizations. The problems of the stability of characterization of the normal distribution by means of identically distributed linear form and monomial are considered in a uniform metric. The moment and symmetry conditions are considerably weaker in comparison to previously published versions of the theorem. The choice of a uniform metric appears to be very natural in the framework of Zolotarev’s methodology advising to investigate stability.