Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator H with a periodic potential perturbed by a function ?W that is periodic in two variables and exponentially decreasing in the third variable. Here, ? is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrödinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ?W. We obtain simple formulas for the transmission and reflection probabilities.     

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.     

The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.     

The three-colour model with domain wall boundary conditions

We study the partition function for the three-colour model with domain wall boundary conditions. We express it in terms of certain special polynomials, which can be constructed recursively. Our method generalizes Kuperbergʼs proof of the alternating sign matrix theorem, replacing the six-vertex model used by Kuperberg with the eight-vertex-solid-on-solid model. As applications, we obtain some combinatorial results on three-colourings. We also conjecture an explicit formula for the free energy of the model.     

Standard model of fermions on the domain wall in five-dimensional space-time

The mechanism of generation of the Standard Model for fermions on the domain wall in five-dimensional space-time is presented. As a result of self-interaction of five-dimensional fermions and gravity induced by matter fields, in the strong coupling regime, in the model there arises a spontaneous translational symmetry breaking, which leads to localization of light particles on a 3 + 1-dimensional domain wall (“3-brane”) that is embedded into a five-dimensional anti-de Sitter space-time (AdS₅). Appropriate low-energy, effective action, classical kink-like vacuum configurations for the gravity and scalar fields are investigated. Mass spectra for light composite particles and their coupling constants interaction in ultra-low-energy, which localize on the brane, are explored. We establish estimates of characteristic scales and constants interactions of the model and also a relation between the bulk five-dimensional gravitational constant, curvature of AdS₅ space-time, and brane Newton’s constants. The induced cosmological constant on the brane exactly vanishes in all orders of the theory perturbation. We find out that scalar interaction is strongly suppressed at ultra-low-energy, and the brane fluctuations (branons) are suitable “sterile” canditates for explanation of the phenomenon of Dark Matter. Bibliography: 21 titles. Dedicated to the 100th birthday of M. P. Bronstein __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 5–29.     

On the domain derivative for scattering by impenetrable obstacles in chiral media

For scattering of electromagnetic waves in a chiral medium bysome perfectly conducting inclusions, we study the dependenceof the scattered field on the boundary of the inclusions andshow its Fréchet differentiability in appropriate spaces.Further, we derive a characterization of the derivative as asolution to some corresponding chiral boundary value problem.Our proof contains a new approach to rigorously derive thischaracterization.     

双连通区域上的电磁波散射问题数值解

对于双连通区域上的电磁波散射问题,通过位势理论将其转化为边界积分方程组问题,然后采用Nystrom法和配置法对其离散求解,针对不同形状的障碍散射体,给出远场模式的数值解.     

Induced gravity and universe creation on the domain wall in five-dimensional space-time

We present a model with a strong fermion self-interaction and an induced gravitational interaction in five-dimensional space-time. In the strong-coupling regime, this model develops a spontaneous breaking of translational invariance, which results in localizing light particles on the (3+1)-dimensional domain wall embedded in the anti-de Sitter five-dimensional space-time. We obtain the corresponding low-energy effective action and investigate kink-type vacuum solutions in a quasiflat Riemannian metric. We discuss the physics of light particles in 3+1 dimensions and establish the fundamental relations between the induced gravitational constant and both the curvature in the five-dimensional anti-de Sitter space-time and the Newton gravitational constant of our universe. Dedicated to Yu. V. Novozhilov on his 80th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 4–22, July, 2006.     

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including V(x):=−γ|x|^−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|^−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.     

An analysis of the BEM-FEM non-overlapping domain decomposition method for a scattering problem

In this paper, we analyze the BEM-FEM non-overlapping domain decomposition method introduced in Boubendir [Techniques de Décomposition de Domaine et Méthode d’Equations Intégrales, Ph.D. Thesis, INSA, Toulouse, France, 2002] and improved in Boubendir et al. [A coupling BEM-FEM method using a FETI-LIKE domain decomposition method, in: Proceedings of Waves 2005, Providence, RI, 2005, pp. 188–190] and Bendali et al. [A FETI-like domain decomposition method for coupling FEM and BEM in large-size problems of acoustic scattering, to appear.] The transmission conditions used in this method introduce a quantity that prevents the approach of Després [Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique, Le théorème de Borg pour l’équation de Hill vectorielle, Ph.D. Thesis, Paris VI University, France, 1991] to establish convergence to be adapted. However, we show that convergence can be established when the geometry allows for a decomposition of the solution into propagating and evanescent portions with a methodology based on modal analysis. Here, we exemplify this in the case of circular cylindrical geometries where the derivations can be based on properties of Bessel functions.     

High-energy bounds on helicity amplitudes in elastic pp scattering in the small-angle domain

Bounds on helicity amplitudes in elastic pp scattering follow from the unitarity and analyticity conditions. We discuss the maximum admissible energy dependence, corresponding to saturating the obtained unitarity restrictions, of the spin parameters of elastic pp scattering. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 473–482, December, 2006.     

Limiting domain wall energy for a problem related to micromagnetics

We study the asymptotic limit of a family of functionals related to the theory of micromagnetics in two dimensions. We prove a compactness result for families of uniformly bounded energy. After studying the corresponding one‐dimensional profiles, we exhibit the Γ‐limit (“wall energy”), which is a variational problem on the folding of solutions of the eikonal equation |∇g| = 1. We prove that the minimal wall energy is twice the perimeter. © 2001 John Wiley & Sons, Inc.     

Indefinite metric and scattering by a domain with a small hole

For the problem of plane waves scattered by a domain with a small hole, we suggest a model based on the theory of self-adjoint extensions of symmetric operators in a space with indefinite metric. For two-dimensional problems of scattering on a line with a hole and on a semi-ellipse connected by a hole with a half-plane, we justify the choice of extension that guarantees the coincidence of the model solution with the solution of the actual problem in the far zone with a high degree of accuracy.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 837–850, December, 1995.The authors are grateful to B. S. Pavlov and L. M. Grigoryan for useful discussion.The work was partially supported by the State Commission on Higher Education of the Russian Federation under grant No. 94-2.7-1067.     

带尖角的障碍声波散射区域的反演

对带尖角的障碍声波散射区域进行了反演,其前提条件是整体场满足奇次Dirichlet边界条件．在用Nystrom方法解正问题的过程中,由于采用等距网格积分给尖角处带来很差的收敛性,这是因为双层位势的积分算子的核在尖角处有Mellin型奇性,不再是紧算子;为此采用梯度网格,数值例子表明该处理方法的有效可靠性．     

Three-coloring statistical model with domain wall boundary conditions: Functional equations

We consider the Baxter three-coloring model with boundary conditions of the domain wall type. In this case, it can be proved that the partition function satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.     

Three-coloring statistical model with domain wall boundary conditions: Trigonometric limit

We consider a nontrivial trigonometric limit of the three-coloring statistical model with the domain wall boundary conditions. In this limit, we solve the previously constructed functional equations and find a new determinant representation for the partial partition functions.     

Slow laminar flow in a converging or diverging channel with suction at one wall and blowing at the other wall

Résumé On établit et résoud l'équation de l'écoulement laminaire dans un canal convergent ou divergent avec aspiration de la couche limite sur l'une des parois et soufflement sur l'autre.