RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering.

The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W ^1,p -functions due to P. Hajlasz, and a modification of the classical Triebel's Maximal Inequality.     

A generalization of the Birman trace theorem

A simple and direct proof is presented of a version of the Birman trace theorem which is sufficiently general to apply to potential scattering with absorption at local singularities.     

Finite-Dimensional Approximations for Scattering Theory Operators in the Wave-Packet Representation

We consider some types of packet discretization for continuous spectra in quantum scattering problems. As we previously showed, this discretization leads to a convenient finite-dimensional (i.e., matrix) approximation for integral operators in the scattering theory and allows reducing the solution of singular integral equations connected with the scattering theory to some suitable purely algebraic equations on an analytic basis. All singularities are explicitly singled out. Our primary emphasis is on realizing the method practically.     

High energy asymptotics of the scattering amplitude for the Schrödinger equation

We find an explicit function approximating at high energies the kernel of the scattering matrix with arbitrary accuracy. Moreover, the same function gives all diagonal singularities of the kernel of the scattering matrix in the angular variables. This paper is dedicated to Jean-Michel Combes on the occasion of his sixtieth birthday.     

Abstract Lax-Phillips scattering scheme for second-order operator-differential equations

We construct an analog of the Lax-Phillips scattering scheme for an abstract operator-differential equation u _u=-Lu under certain restrictions imposed on the operator L. In particular, we construct the incoming and outcoming subspaces and describe singularities of the scattering matrix in terms of the spaces of boundary values.     

A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities

We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.     

Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

We prove that the singularities of a complex valued potential q in the Schrödinger hamiltonian Δ+q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for ${q \in W^{\alpha,2}}We prove that the singularities of a complex valued potential q in the Schr?dinger hamiltonian Δ+q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for q ? W^a,2{q \in W^{\alpha,2}} for α ≥ 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is ${\epsilon > 1/2}${\epsilon > 1/2}, increasing with α. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.     

Caustics of interior scattering

The paper is devoted to geometrical optics of short linear waves in an inhomogeneous anisotropic medium. We find some typical singularities of caustics that arise due to the so-called interior scattering of waves, the mathematical theory of which was developed by V.I. Arnold in 1988.     

Composition of Fourier Integral Operators with Fold and Blowdown Singularities

ABSTRACT

The purpose of this work is to present results about the composition of Fourier integral operators with certain singularities, for which the composition is not again a Fourier integral operator. The singularities considered here are folds and blowdowns. We prove that for such operators, the Schwartz kernel of F*F belongs to a class of distributions associated to two cleanly intersection Lagrangians. Such Fourier integral operators appear in integral geometry, inverse acoustic scattering theory and Synthetic Aperture Radar imaging, where the composition calculus can be used as a tool for finding approximate inversion formulas and for recovering images.     

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

The current work sets forth a practical approach to numerically solve two‐dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite‐difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross‐section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Singularities and the Rayleigh Hypothesis for Solutions to the Helmholtz Equation

Possible and actual singularities in the analytic continuationof the solution to the Dirichlet problem for the two-dimensionalHelmholtz equation are studied in order to investigate the Rayleighhypothesis of scattering theory. The procedure uses a representationfor the solution to the analytic Cauchy problem and relatesone type of singularity in the Schwarz function associated withthe boundary curve to singularities in the solution. The resultsprovide confiration of criteria for the validity of the Rayleighhypothesis that have been given by van den Berg and Fokkema.     

Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

ABSTRACT

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show that existence of solutions heavily depends on the strength and the location of the singularities. We associate to the problem the corresponding Rayleigh quotient and give both sufficient and necessary conditions on masses and location of singularities for the minimum to be achieved. Both the cases of whole ? ^N and bounded domains are taken into account.     

Microlocal Analysis of Synthetic Aperture Radar Imaging

We consider Synthetic Aperture Radar (SAR) in which backscattered waves are measured from locations along a single flight path of an aircraft. Emphasis is on the case where it is not possible to form a beam with the radar. The article uses a scalar linearized mathematical model of scattering, based on the wave equation. This leads to a forward (scattering) operator, which maps singularities in the coefficient of the wave equation (viewed as a singular perturbation about a constant coefficient) to singularities in the scattered wave field. The goal of SAR is to recover a picture of the singular support of the coefficient, i.e., an a image of the underlying terrain. Traditionally, images are produced by backprojecting the data. This is done by applying the adjoint of the scattering operator to the data. This backprojected image is equivalent to that obtained by applying to the perturbed coefficient the composition of the scattering operator followed by its adjoint. We analyze this composite operator, and show that it is a paired Lagrangian operator. The properties of such operators explain the origin of certain artifacts in the backprojected image.     

The spectrum and the scattering problem for the Schrödinger operator in a magnetic field

The main result of this paper is the proof of a nonexistence theorem for solutions with nonzero real singularities to the problem of scattering theory for the Schrödinger operator with magnetic and electric potentials.     

Riemann-Hilbert approach to TD equation with nonzero boundary condition

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

A least-squares finite element method for solving the polygonal-line arc-scattering problem

This paper is concerned with the scattering problem of a polygonal-line arc. We solve this polygonal-line arc-scattering problem by a least-squares finite element method. In the method, Fourier–Bessel functions is used to capture the singularities around tips and corners. A combination of fundamental solutions is used to represent the scattered field towards infinity. We also analyse the convergence and give an error estimate of the method. Numerical experiments are also presented to show the effectiveness of our method.     

Uniform coordinates and spectral asymptotics for solutions of the scattering problem for the Schrödinger equation

Uniform asymptotic formulas at large distances and for a large value of the spectral parameter are obtained for solutions of the scattering problem. The singularities of the scattering amplitude in the angular variables and its asymptotic expression in the spectral parameter are described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 171–199, 1977.     

Characterization of Scattering Data for the AKNS System

We characterize the scattering data of the AKNS system with vanishing boundary conditions. We prove a 1,1-correspondence between L ¹-potentials without spectral singularities and Marchenko integral kernels which are sums of an L ¹ function (having a reflection coefficient as its Fourier transform) and a finite exponential sum encoding bound states and norming constants. We give characterization results in the focusing and defocusing cases separately.