**Description:**

The
inverse obstacle scattering problem consists of finding the unknown surface of
a body (obstacle) from the scattering *A*(ß;a;k), where *A*(ß;a;*k)*
is the scattering amplitude, ß;a ? *S*^{2}^{
}is the direction of the scattered, incident wave, respectively, *S*^{2}
is the unit sphere in the R^{3 }and k > 0 is the modulus of the wave
vector. The scattering data is called non-over-determined if its dimensionality
is the same as the one of the unknown object. By the dimensionality one
understands the minimal number of variables of a function describing the data
or an object. In an inverse obstacle scattering problem this number is 2, and
an example of non-over-determined data is *A*(ß) := *A*(ß;a_{0};,*k*_{0}).
By sub-index 0 a fixed value of a variable is denoted.

It
is proved in this book that the data *A*(ß), known for all ß in an open
subset of *S*^{2}, determines uniquely the surface *S *and
the boundary condition on *S*. This condition can be the Dirichlet, or the
Neumann, or the impedance type.

The
above uniqueness theorem is of principal importance because the
non-over-determined data are the minimal data determining uniquely the unknown *S.
*There were no such results in the literature, therefore the need for this
book arose. This book contains a self-contained proof of the existence and
uniqueness of the scattering solution for rough surfaces.

Contents:

**Preface**

**Chapter1. Introduction
**

**Chapter 2. The Direct
Scattering Problem** • Statement of the Problem • Uniqueness of
the Scattering Solution • Existence of the Scattering Solution • Properties of
the Scattering Amplitude

**Chapter 3. Inverse
Obstacle Scattering** • Statement of the Problem • Uniqueness of
the Solution to Obstacle Inverse Scattering Problem with the Data *A*(*ß
*) • Uniqueness of the Solution to the Inverse Obstacle Scattering Problem
with Fixed-Energy Data • Uniqueness of the Solution to Inverse Obstacle
Scattering Problem with Non-Over-Determined Data • Numerical Solution of the
Inverse Obstacle Scattering Problem with Non-Over-Determined Data

**A. Existence and
Uniqueness of the Scattering Solutions in the Exterior of Rough Domains** •
Introduction • Notations and Assumptions • Function Spaces • Statement of the
Problem • The Weak Formulation of the Scattering Problem • Uniqueness Theorem •
Existence of the Scattering Solutions for Compactly Supported Potentials •
Existence for the Equation with the Absorption • The Limiting Absorption
Principle • Existence of the Scattering Solution for Decaying Potentials •
Existence for the Equation with Absorption • The Limiting Absorption Principle

**Bibliography **

**Author’s Biography**

About the Author:

**Alexander G. Ramm, **Ph.D., was born in
Russia, immigrated to the U.S. in 1979, and is a U.S. citizen. He is Professor
of Mathematics with broad interests in analysis, scattering theory, inverse
problems, theoretical physics, engineering, signal estimation, tomography,
theoretical numerical analysis, and applied mathematics. He is an author of 690
research papers, 16 monographs, and an editor of 3 books. He has lectured in
many universities throughout the world, presented approximately 150 invited and
plenary talks at various conferences, and has supervised 11 Ph.D. students. He
was Fulbright Research Professor in Israel and in Ukraine, distinguished
visiting professor in Mexico and Egypt, Mercator professor, invited plenary speaker
at the 7th PACOM, won the Khwarizmi international award, and received other
honors. Recently he solved inverse scattering problems with non-over-determined
data and the many-body wave-scattering problem when the scatterers are small
particles of an arbitrary shape; Dr. Ramm used this theory to give a recipe for
creating materials with a desired refraction coefficient. He gave a solution to
the refined Pompeiu problem, and proved the refined Schiffer’s conjecture.

Target Audience:

This
book contains a self-contained proof of the existence and uniqueness of the
scattering solution for rough surfaces. This book is useful for people
interested in numerical analysis, statistics and computer science.