Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g., in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory, reducing the problem of resonant wave–particle interaction to the analysis of slow passages of a particle through a resonance. We discuss in detail two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants and chaotization of particles; they also may lead to significant (almost free) acceleration of particles and may govern transport in the phase space. We calculate the characteristic times of mixing due to resonant effects and separatrix crossings, and discuss the relative importance of these phenomena.     

The inverse scattering problem for a perturbed difference Hill equation

We consider the inverse scattering problem for the difference analog of a perturbed Hill equation. The perturbation coefficients are recovered from the periodic coefficients and from the scattering data.     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

On the Inverse Problem for Scattering of Electromagnetic Radiation by a Periodic Structure

We consider a smooth perturbation  δε( x , y , z )  of a constant background permittivity  ε=ε₀  that varies periodically with x , does not depend on y , and is supported on a finite-length interval in z . We investigate the theoretical and numerical determination of such perturbation from (several) fixed frequency y -invariant electromagnetic waves.
By varying the direction and frequency of the probing radiation a scattering matrix is defined. By using an invariant-imbedding technique we derive an operator Riccati equation for such scattering matrix. We obtain a theoretical uniqueness result for the problem of determining the perturbation from the scattering matrix.
We also investigate a numerical method for performing such reconstruction using multi-frequency information of the truncated scattering matrix. This relies on ideas of regularization and recursive linearization. Numerical experiments are presented validating such approach.     

Aharonov�CBohm Effect in Resonances of Magnetic Schr?dinger Operators with Potentials with Supports at Large Separation

Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov–Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field.     

A finite element method for approximating electromagnetic scattering from a conducting object

We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell's equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincaré-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.     

Description of pair potentials for which the scattering in a quantum system of three one-dimensional particles is free of diffraction effects

The coordinate asymptotics of the solution of the scattering problem for a system of three one-dimensional particles contains, besides plane and spherical waves, also Fresnel waves which arise also in the two-dimensional problem of a plane wave on a semiinfinite screen. One describes explicitly the class of potentials for which the Fresnel waves do not occur in the coordinate asymptotics. This class is somewhat wider than the class of nonrefleeting potentials.     

Decay and scattering of small solutions for Rosenau equations

We study the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. For a class of nonlinearity of the perturbation, the global small solution was obtained, and the decay and scattering for small amplitude solution are established.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

Asymptotic Solution of A Multichannel Scattering Problem with A Nonadiabatic Coupling

We consider a multichannel scattering problem in an adiabatic representation. We assume that the nonadiabatic coupling matrix has a nontrivial value at large internuclear separations, and we construct asymptotic solutions at large internuclear distances. We show that these solutions up to the first order of the perturbation theory are identical to the asymptotic solutions of the reprojection approach, which was previously proposed as a means for solving the electron translation problem in the context of the Born–Oppenheimer method.     

New Formulae for the Wave Operators for a Rank One Interaction

We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson’s theorem for this model.     

Computation of Feynman loop integrals

We address multivariate integration and extrapolation techniques for the computation of Feynman loop integrals. Loop integrals are required for perturbation calculations in high energy physics, as they contribute corrections to the scattering amplitude and the cross section for the collision of elementary particles. We use iterated integration to calculate the multivariate integrals. The combined integration and extrapolation methods aim for an automatic calculation, where little or no analytic manipulation is required before the numeric approximation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.     

Mathematical Model of Resonances and Tunneling in a System with a Bound State

We study the asymptotic behavior of the residue at the pole of the analytic continuation of the scattering matrix as the imaginary part of the pole tends to zero in the case where the phase space of a quantum mechanical system is a direct sum of two spaces and the nonperturbed evolution operator reduces each of these spaces and has a discrete spectrum in one of them and a continuous spectrum in the other. The perturbation operator mixes the subspaces and generates a resonance. We prove that under certain symmetry conditions in such a system, the scattering amplitude changes sharply in a neighborhood of the real part of the pole of the scattering matrix, and the system demonstrates tunneling or a resonance of the scattering amplitude.     

Transformation Operator for Jacobi Matrices with Asymptotically Periodic Coefficients

We construct the transformation operator for the scattering problem with a periodic background under the assumption that the coefficients of the perturbation have a first finite moment. By means of the Marchenko approach [Marchenko, V. (1986) Sturm–Liouville Operators and Applications. Birkhäuser, Basel, Switzerland] we derive an estimate on the kernel of this transformation operator that allow us to study the inverse problem solution in the prescribed class of perturbations.     

Direct and inverse multichannel scattering problems

We consider small oscillations of a system of pairwise interacting particles in an external field near a stable equilibrium. The system is assumed to consist of finitely many channels, i.e., semi-infinite linear chains of particles, attached to a scatterer, which is a finite system of interacting particles. Direct and inverse scattering problems are considered. In particular, an algorithm finding characteristics of the channels on the basis of scattering data is given.     

The virtual mass of a sphere in a suspension of spherical particles

The problem of the virtual mass of a sphere, moving in an ideal incompressible fluid when there are other identical spherical particles of arbitrary mass present is considered. A solution is constructed for the velocity potential of the fluid in the form of the superposition of perturbation fields, introduced into the flow by each of the particles. The perturbation fields are obtained in the form of functional series, the coefficients of which are mutually consistent by a defined system of equations. An explicit expression is obtained for the hydrodynamic force acting on the sphere in the form of a function of the coordinates of all the particles. A simple analytical dependence of the mean value of the force and the virtual mass of the sphere on the particle-to-fluid density ratio in a first approximation of the volume fraction of the dispersed phase is obtained for a statistically uniform distribution of the dispersed particles in the suspension, using the procedure of averaging over their different possible configurations in space.     

关于二层海中的二阶波浪绕射问题*

在本文中,我们用二层海模型,探讨了层化海洋中任意三维物体的二阶波浪绕射问题,给出了多色波场中二阶波浪散射势边值问题的数学提法以及基于一个弱的远场辐射条件下解的表式。同时,利用Green定理,并通过引入一个辅助势函数,我们导出了结构所受二阶波浪荷载的积分表式。结果表明,海水的层化特性对结构物所受之二阶差频波浪荷载可能具有显著的影响。     

An inverse scattering problem for a perturbed Hill's operator

We consider the inverse scattering problem for the operator L=?d²/dx²+p(x)+q(x), x ∈ R¹. The perturbation potential q is expressed in terms of the periodic potential p and the scattering data. We also obtain identities for the eigenfunctions of the unperturbed Hill's operator L₀=?d²/dx²+p(x).