Vol 238, No 5 (2019)
- Year: 2019
- Articles: 16
- URL: https://journals.rcsi.science/1072-3374/issue/view/15000
Article
On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion
Abstract
The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrödinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrödinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.
Local Boundary Controllability in Classes of Differentiable Functions for the Wave Equation
Abstract
The well-known fact following from the Holmgren-John-Tataru uniqueness theorem is a local approximate boundary L2-controllability of the dynamical system governed by the wave equation. Generalizing this result, we establish the controllability in certain classes of differentiable functions in the domains filled up with waves.
Some Aspects of the Scattering Problem for a System of Three Charged Particles
Abstract
The question of influence of the spectral neighborhood of an accumulative point of bound energies of a pair subsystem on the structure of eigenfunctions of the continuous spectrum for a system of three charged quantum particles is studied. The unified contribution of pair high-excited states are separated in the coordinate asymptotics of such functions.
Modelling Equation of Electromagnetic Scattering on Thin Dielectric Structures
Abstract
In this research, we study the scattering of electromagnetic waves by a dielectric impediment in 2D geometry. The impediment is determined by an inhomogeneous component of the refractive index in the Helmholtz equation. It is assumed that the characteristic gauge of one of the two impediment sizes is much lesser than the length of waves generated by a monochromatic point source. Nevertheless, the structure of the impediment is taken into consideration in the process of calculating the scattered field. The scattered field is defined by a derived model integral equation the unique solvability of which is proved.
Justification of a Wavelet-Based Integral Formula for Solutions of the Wave Equation
Abstract
An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed from one solution by means of transformations of shift, scaling, and the Lorentz transform. Sufficient conditions are derived, which ensure the pointwise convergence of the obtained improper integral in the space of parameters. The convergence of this integral in ℒ2 norm is proved as well. Bibliography: 22 titles.
Comparison of Asymptotic and Numerical Approaches to the Study of the Resonant Tunneling in Two-Dimensional Symmetric Quantum Waveguides of Variable Cross-Sections
Abstract
The waveguide considered coincides with a strip having two narrows of width ε. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as ε → 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of ε, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles.
Weak Solutions of Hopf Type to 2D Maxwell Flows with Infinite Number of Relaxation Times
Abstract
A system of equations describing the motion of fluids of Maxwell type is considered:
Here K(t) is an exponential series \( K(t)=\sum \limits_{s=1}^{\infty }{\beta}_s{e}^{-{\alpha}_st} \). The existence of a weak solution for the initial boundary value problem
is proved.
Leontovich–Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution
Abstract
This paper continues a series of publications on the shortwave diffraction of the plane wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock’s main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping waves in the boundary layer of the scatterer in the shadow zone.
Model of a Saccular Aneurysm of the Bifurcation Node of an Artery
Abstract
Modified Kirchhoff transmission conditions in a simple one-dimensional model of a branching artery developed by the authors, allow one to describe an anomaly of its bifurcation node, congenital or acquired due to trauma or disease of a vessel wall. The pathology of the blood flow through the damaged node and the methods of determining the aneurysm parameters from the data measured at the peripheral parts of the circulatory system by solving inverse problems are discussed.
The Wave Field Near a Narrow Convex Impedance Cone Completely Illuminated by a Plane Incident Wave
Abstract
An acoustic incident plane wave completely illuminates a narrow convex cone satisfying the impedance boundary condition on its surface. The wave field at far distances from the vertex of the cone and in some close neighborhood of the cone’s surface is asymptotically computed.
On an Inverse Dynamic Problem for the Wave Equation with a Potential on a Real Line
Abstract
The inverse dynamic problem for the wave equation with a potential on a real line is considered. The forward initial-boundary value problem is set up with the help of boundary triplets. As an inverse data, an analog of the response operator (dynamic Dirichlet-to-Neumann map) is used. Equations of the inverse problem are derived; also, a relationship between the dynamic inverse problem and the spectral inverse problem from a matrix-valued measure is pointed out.
Scattering Amplitudes in a Neighborhood of Limit Rays in Short-Wave Diffraction by Elongated Bodies of Revolution
Abstract
In the paper, diffraction problems of a plane wave by smooth, convex, and elongated bodies of revolution are considered within the framework of short-wave approximation (axially symmetric cases). The scattering amplitudes are calculated in the direction of limit rays, and the influence of the elongation of the scatterers on the amplitudes behavior is investigated. Mathematical techniques of our approach are based on the Green’s formulas in the exterior of the scatterers and numerical calculations of the wave field current in the boundary layers in the vicinity of the light-shadow zone. It is established that the elongation of axially symmetric bodies relatively weakly affects the scattering amplitudes of the short-wave asymptotics. The main contribution to the amplitudes is made by the solution of the 2D diffraction problem by a convex, smooth curve in the cross section of the scatterers by a plane containing the rotation axis.
To the Calculations of Scattering Amplitudes in Diffraction Problems for Elongated Bodies of Revolution
Abstract
This paper is a complement to the article “Scattering amplitudes in a neighborhood of limit rays in short-wave diffraction by elongated bodies of revolution”. It contains discussions of some points of the article, which worth of more detailed considerations, such as the influence of the integration limits on the computation result of scattering amplitudes and the estimation of permissible values of scattering angle intervals as functions of parameters of the problems.
Asymptotics of the Resonant Tunneling of High-Energy Electrons in Two-Dimensional Quantum Waveguides of Variable Cross-Section
Abstract
A waveguide occupies a strip in ℝ2 having two identical narrows of small diameter ε. An electron wave function satisfies the Helmholtz equation with the homogeneous Dirichlet boundary condition. The energy of electrons may be rather high, i.e., any (fixed) number of waves can propagate in the strip far from the narrows. As ε → 0, a neighborhood of a narrow is assumed to transform into a neighborhood of the common vertex of two vertical angles. The part of the waveguide between the narrows as ε = 0 is called the resonator. An asymptotics of the transmission coefficient is obtained in the waveguide as ε → 0. Near a degenerate eigenvalue of the resonator, the leading term of the asymptotics has two sharp peaks. Positions and shapes of the resonant peaks are described.
On Minimal Entire Solutions of the One-Dimensional Difference Schrödinger Equation with the Potential υ(z) = e−2πiz
Abstract
Let z ∈ ℂ be a complex variable, and let h ∈ (0, 1) and p ∈ ℂ be parameters. For the equation ψ(z + h) + ψ(z − h) + e−2πizψ(z) = 2 cos(2πp)ψ(z), solutions having the minimal possible growth simultaneously as Im z → ∞ and as Im z → − ∞ are studied. In particular, it is shown that they satisfy one more difference equation ψ(z + 1) + ψ(z − 1) + e−2πiz/hψ(z) = 2 cos(2πp/h)ψ(z).