Alexander G. Ramm

Multidimensional inverse scattering problems

Summary form only given, as follows. An overview of the authors results is given. The inverse problems for obstacle, geophysical and potential scattering are considered. The basic method for proving uniqueness theorems in one- and multi-dimensional inverse problems is discussed and illustrated by numerous examples. The method is based on property C for pairs of differential operators. Property C stands for completeness of the sets of products of solutions to homogeneous differential equations. To prove a uniqueness theorem in the inverse scattering problem one assumes that there are two operators which generate the same scattering data. This assumption allows one to derive an orthogonality relation from which, via property C, the uniqueness theorem follows. New results are discussed. These include property C for ordinary differential equations, inversion of I-function (impedance function), inversion of incomplete scattering data (for example, the phase shift of s-wave without the knowledge of bound states and norming constants but assuming a priori that the potential has compact support, etc). Analytical solution of the ground-penetrating radar problem is outlined. Open problems are formulated.

On stable numerical differentiation

Based on a regularized Volterra equation, two different approaches for numeri- cal differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete fa- vorably with the variational regularization method for stable calculating the derivatives of noisy functions.

Theory and applications of some new classes of integral equations

We may not be able to make you love reading, but theory and applications of some new classes of integral equations will lead you to love reading starting from now. Book is the window to open the new world. The world that you want is in the better stage and level. World will always guide you to even the prestige stage of the life. You know, this is some of how reading will give you the kindness. In this case, more books you read more knowledge you know, but it can mean also the bore is full.

Wave Scattering by Small Bodies of Arbitrary Shapes

This article reviews Wave Scattering by Small Bodies of Arbitrary Shapes by Alexander G. Ramm , Hackensack, 2005. xviii+293 pp.

Multidimensional inverse problems and completeness of the products of solutions to PDE

66 (hardcover), ISBN: 9812561862.

Many-body wave scattering by small bodies and applications

A method is given for proving uniqueness theorems for some inverse problems. The method is based on a result on completeness of the products of solutions to PDE. As an example, the following uniqueness theorems are proved: (1) the scattering amplitude A(θ′, θ, k) known for all θ′, θ ϵ S2 and a fixed k > 0 determines the compactly supported q(x) ϵ L2(D) uniquely; (2) the surface data u(x, y, k) known for all x, y ϵ P:= {x:x3 = 0} and a fixed k > 0 determine the compactly supported ν(x) ϵ L2(D), D ⊂ R−3:= {x:x3 0 is arbitrarily small; determine aj(x), j = 1, 2, uniquely. Here ▽2u + k2u + k2a1x) + ▽ · (a2(x) ▽u) = −δ(x − y) in R3, a1 ϵ L2(D), a2 ϵ H2(D); the same conclusion holds if the surface data are known at two distinct frequencies. (4) The surface data u(x, y, k) known for all x, y ϵ P and all k > 0 determine v(x) and h(k) uniquely. Here [▽2 + k2 + k2ν(x)] u = −δ(x − y) h(k), ν(x) ϵ L2(D), h(k) is Fourier transform of a wavelet of compact support; (5) the conductivity σ(x)ϵ W2,2(D), σ(x) ⩾ c > 0, is uniquely determined by the measurements of u and σuN on ∂D. Here N is the outward normal to ∂D, D ⊂ R3 is a bounded domain with a smooth boundary ∂D, ▽ · (σ(x) ▽u) = 0 in D. (6) Necessary and sufficient conditions are given for a function A(θ′, θ, k), θ′, θ ϵ S2, k > 0 is fixed, to be the scattering amplitude corresponding to a local potential from a certain class.

Spectral and scattering theory

A rigorous reduction of the many-body wave scattering problem to solving a linear algebraic system is given bypassing solving the usual system of integral equation. The limiting case of infinitely many small particles embedded into a medium is considered and the limiting equation for the field in the medium is derived. The impedance boundary conditions are imposed on the boundaries of small bodies. The case of Neumann boundary conditions (acoustically hard particles) is also considered. Applications to creating materials with a desired refraction coefficient are given. It is proved that by embedding a suitable number of small particles per unit volume of the original material with suitable boundary impedances, one can create a new material with any desired refraction coefficient. The governing equation is a scalar Helmholtz equation, which one obtains by Fourier transforming the wave equation.

A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators

Wave Scattering in 1-D Nonconservative Media T. Aktosun, et al. Resolvent Estimates for Schrodinger-type and Maxwell Equations with Applications M. Ben-Artzi, J. Nemirovsky. Symmetric Solutions of Ginzburg-Landau Equations S. Gustafson. Quantum Mechanics and Relativity: Their Unification by Local Time H. Kitada. On Embedded Eigenvalues of Perturbed Periodic Schrodinger Operators P. Kuchment, B. Vainberg. On Principal Eigenvalues for Indefinite-Weight Elliptic Problems Y. Pinchover. Scattering by Obstacles in Acoustic Waveguides A.G. Ramm, G.N. Makrakis. Recovery of Compactly Supported Spherically Symmetric Potentials from the Phase Shift of the s-Wave A.G. Ramm. A Turning Point Problem Arising in Connection with a Limiting Absorption Principle for Schrodinger Operators with Generalized Von Neumann-Wigner Potentials P. Rejto, M. Taboada. Eigenvalue Problems for Semilinear Equations M. Schechter. Spectral Operators Generated by 3-Dimensional Damped Wave Equation and Applications to Control Theory M.A. Shubov. Invertibility of Nonlinear Operators and Parameter Continuation Method V.A. Trenogin. Sturm-Liouville Differential Operators with Singularities V.Yurko. Index.

Stability estimates in inverse scattering

(2001). A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators. The American Mathematical Monthly: Vol. 108, No. 9, pp. 855-860.

Completeness of the products of solutions of PDE and inverse problems

An algorithm is given for calculating the solution to the 3D inverse scattering problem with noisy discrete fixed energy data. The error estimates for the calculated solution are derived. The methods developed are of a general nature and can be used in many applications: in nondestructive evaluation and remote sensing, in geophysical exploration, medical diagnostics, and technology.