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.     

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.     

Electromagnetic field and dispersion characteristic analysis of absorbing onion-like carbon tube waveguides

Here, we present our calculation results of the electromagnetic field distributions and the dispersion characteristics of open cylindrical tube waveguides. The analyzed waveguides are made of the onion-like carbon (OLC) material. The solution of the boundary problem was fulfilled by the partial area method (Nickelson et al. in electrodynamical analysis of open lossy metamaterial waveguide and scattering structures. InTech, UK, pp 27–58 [1]). We have determined the complex roots of the dispersion equation using of the Muller method. It discovered the very complicated dependencies of the phase and attenuation constants on the waveguide radii. Such dependencies arise because the OLC material is the highly dispersive and absorbing one. We have investigated the high-frequency cutoff frequency of the propagating hybrid modes HE₁₁ and HE₁₂ dependent on the tube waveguide external and internal radii. We found that it is possible to reach the one-mode regime of OLC tube waveguide.     

Truss optimization under stiffness, stability constraints and random loading

An actual design of light-weight structures must evaluate strength, stiffness and stability constraints as well as the nature of external loading. A designed structure must satisfy optimality and safety criterions per prescribed maintenance period. One faces the known difficulties when trying to implement several from the above mentioned requirements into optimization problem for further successful numerical realization. A method to formulate the optimization problem, incorporating all above described criterions, the mathematical model and algorithm to solve it numerically, taking into account the stochastic nature of external loading, are presented for elastic–plastic truss-type structure.     

On the Stability of Eaton’s Characterization by the Properties of Linear Forms

We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics $(X_{1}+\cdots +X_{k_{1}})/k_{1}^{1/\alpha}We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics and have the same distribution as the monomial X ₁, then this monomial has a symmetric stable distribution of order α. The stability estimation in this theorem is investigated in the λ ₀-metric.      

Microwave scattering diagrams of three-layered SiC-metamaterial/gyrotropic ferrite-SiC cylinders

We present here for the first time the rigorous solution of the boundary diffraction problem of microwave scattering by a multilayered 2D cylinder. The cylinder layers may be made of isotropic, uniaxial anisotropic, electrically and (or) magnetically gyrotropic materials. The number and thickness of the layers may have arbitrary values in our solution. We calculated scattering diagrams (a radial component of real part of the Poynting vector) inside and outside of cylinder using the solution. Here we present scattering diagrams from a three-layered cylinder made of SiC and metamaterial or saturate magnetized ferrite. Diagrams were computed for wave incidence angles θ=π/2,π/3,π/6 inside of metamaterial/ferrite layer at a distance of 1 mm and outside of cylinder at a distance of 2.5 mm from the cylinder axis.