Construction of scattering operators by the method of binary Darboux transformations

By using the binary Darboux transformations, we construct scattering operators for a Dirac system with special potential depending on 2n arbitrary functions of a single variable. It is shown that one of the operators coincides with the scattering operator obtained by Nyzhnyk in the case of degenerate scattering data. It is also demonstrated that the scattering operator for the Dirac system is either obtained as a composition of three Darboux self-transformations or factorized by two operators of binary transformations of special form. We also consider several cases of reduction of these operators. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1097–1115, August, 2006.     

The scattering of sound waves by an obstacle

In this paper we study the scattering of acoustic waves by an obstacle ??. We establish the following relation between the scattering kernel S(s, θ, ω) and the support function h_?? of the obstacle: The right endpoint of the support of S(s, θ, ω) as function of s is h_??(θ-ω); h_?? is defined by For Dirichlet boundary condition the result is proved in full generality, for Neumann condition only for backscattering, i.e., for θ = -ω. Since the convex hull of ?? can be recovered from knowledge of h_??, the above result may be useful in reconstructing ?? from scattering data.     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

Microlocal properties of scattering matrices

We consider the scattering theory for a pair of operators H₀ and H = H₀ + V on L²(M, m), where M is a Riemannian manifold, H₀ is a multiplication operator on M, and V is a pseudodifferential operator of order ? μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.     

Small angle scattering andX-ray transform in classical mechanics

We consider the Newton equation

Scattering Frequencies and a Class of Perturbed Systems of Elastic Waves

We consider the system of elastic waves in three dimensions under the presence of an impurity of the medium which we represent by a real-valued function q(x) (or q(x,t)). The medium is assumed to be isotropic and occupies the whole space Ω = ℝ³. We study the location of the scattering frequencies associated with such phenomenon. We conclude that there is a large region on the complex plane which is free of scattering frequencies. In the remaining region they are discrete provided that q satisfies suitable assumptions concerning its behaviour at infinity.     

Estimates for the Scattering Map Associated with a Two-Dimensional First-Order System

Summary. We consider the scattering transform for the first-order system in the plane, We show that the scattering map is Lipschitz continuous on a neighborhood of zero in L ² . Received September 11, 2000; accepted August 27, 2001 Online publication November 5, 2001     

Mathematical basis of the linear sampling method for partially coated obstacles

We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method.     

Characterization of the scattering data for the Sturm–Liouville operator

This work studies the scattering problem on the real axis for the Sturm–Liouville equation with discontinuous leading coefficient and the real‐valued steplike potential q(x) that has different constant asymptotes as x → ± ∞ . We investigate the properties of the scattering data, obtain the main integral equations of the inverse scattering problem, and also give necessary and sufficient conditions characterizing the scattering data. Copyright © 2013 John Wiley & Sons, Ltd.     

Weighted scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant det(I ? U(λ)) of the weighted bond scattering matrix U(λ) of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express some determinant of the weighted bond scattering matrix of a regular covering of G by means of its L-functions.     

Numerical solution of integral equations of first and second kind in scalar and vector problems of diffraction on a cube

We consider the problem of numerical simulation of the scattering of acoustic and electromagnetic waves on a cube whose edge ha s length up to 8 wave lengths of the incident wave. We describe a scheme using a representation of the boundary integral equation in the form of an operator convolution equation on the symmetry group of the cube. We compare the results of numerical solution of integral equations of first and second kind for scalar and vector problems of diffraction of a plane wave on a cube. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 36–45.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete L ₂-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.     

Escape distribution for an inclined billiard

Hénon [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hill’s problem.     

Integrable and nonintegrable cases of the Lax equations with a source

The Korteweg-de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that, depending on the choice of the basis of the eigenfunctions, we have the following three possibilities: (1) evolution equations for the scattering data are nonintegrable; (2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg-de Vries equation with a source at somet>t _o leaves the considered class of functions decreasing rapidly enough asx±; (3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg-de vries equation with a source exists at allt>t _o. All these possibilities are widespread and occur in other Lax equations with a source.Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research 141980 Dubna, Moscow Region, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 471–477, June 1994.     

The n-centre problem of celestial mechanics for large energies

We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations. Received October 16, 2000 / final version received June 18, 2001?Published online August 15, 2001     

Time Evolution of the Scattering Data for the Forced Toda Lattice

Determination of the time evolution of the scattering data for an inverse scattering transform solution of the forced Toda lattice appears to require an overspecification of the boundary condition at the end of the lattice. This appears in the form of an apparent need to specify the values of two functions at the boundary rather than one. We present three different approaches to the resolution of this problem. One approach gives the Maclaurin series (in time) for the scattering data. The second approach gives the scattering data in terms of the solution to a nonlinear, nonlocal partial differential equation. The third approach gives the scattering data in terms of the solution to a linear integral equation. All three approaches reduce to one the number of functions which must be specified to determine a solution. The advantages and limitations of each approach are discussed.     

Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs

We prove that for all an interval graph is ‐Hamilton‐connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best‐known time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k‐Hamilton‐connected can be computed in time.