On the inverse problem for three-dimensional potential scattering

We consider a linear perturbation for the wave equation $\text{□u = 0}\text{in}\text{Ω = E}{}^3$ by “repulsive” smooth potentials q(y) that are small at infinity and suitably small (in magnitude). We use a time-dependent approach to prove that the scattering operator S(q) determines uniquenes uniquely the scatterer q (at least in this class). Energy inequalities will play a central role in our discussion.     

The inverse scattering problem on a half-line with a non-seleconjugate potential matrix

Ukrainian Mathematical Journal -     

Inverse potential scattering problem and its application to the Na-He system

We use the phase-integral approximation in the frame of the inverse scattering theory to reconstruct a potential of the type v = v ₁ + v ₂ in which one component v ₁ is assumed to be known a priori. We introduce an auxiliary potential with two adjustable parameters and show that the unattainable potential range (i.e., the range in which the potential cannot be reconstructed using this method) can be significantly reduced. An excellent agreement is obtained between the original potential and the current results in almost the entire range.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 556–568, March, 2005.     

The problem of scattering by a partially coated crack

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc in R²$R^2$ as the cross section. We assume that the arc is divided into two parts, and one of the two parts is (possibly) coated on one side by a material with surface impedance λ$\lambda$. Applying potential theory, the problem can be reformulated as a boundary integral system. We obtain the existence and uniqueness of a solution to the system by using Fredholm theory.     

Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in H¹ and L²- norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.     

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.     

Homogenization in the scattering problem

The scattering problem is studied, which is described by the equation (-Δ _x +q(x,x/ɛ)−E)ψ = f(x), where ψ = ψ (x,ɛ) ∈ ℂ, x ℂ ℝ ^d , ɛ > 0, E > 0, the function q(x,y) is periodic with respect to y, and the function f is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior as ɛ → O is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential

Expansion of a solution of the Schrödinger equation in terms of solutions of a problem of potential scattering

The Schrödinger equation in three-dimensional space with a Povzner potential is considered. Conditions are found that are necessary so that a solution can be represented in the form where is a solution of a problem of potential scattering. It is proved that the conditions found are also sufficient under more stringent conditions on the potential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im., V. A. Steklova AN SSSR, Vol. 89, pp. 204–209, 1979.     

Phase transitions in relativistic models induced by a stochastic potential

Kharkov State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 1, pp. 49–62, April, 1989.     

Error analysis for a potential problem on locally refined grids

Summary. For the simulation of biomolecular systems in an aqueous solvent a continuum model is often used for the solvent. The accurate evaluation of the so-called solvation energy coming from the electrostatic interaction between the solute and the surrounding water molecules is the main issue in this paper. In these simulations, we deal with a potential problem with jumping coefficients and with a known boundary condition at infinity. One of the advanced ways to solve the problem is to use a multigrid method on locally refined grids around the solute molecule. In this paper, we focus on the error analysis of the numerical solution and the numerical solvation energy obtained on the locally refined grids. Based on a rigorous error analysis via a discrete approximation of the Greens function, we show how to construct the composite grid, to discretize the discontinuity of the diffusion coefficient and to interpolate the solutions at interfaces between the fine and coarse grids. The error analysis developed is confirmed by numerical experiments. Received June 25, 1998 / Revised version received July 14, 1999 / Published online June 8, 2000