Electronic ISBN:  9781470405397 
Product Code:  MEMO/199/933.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 199; 2009; 76 ppMSC: Primary 58; 54; 14; Secondary 37; 46; 20;
This work deals with scattering by obstacles which are finite disjoint unions of strictly convex bodies with smooth boundaries in an odd dimensional Euclidean space. The class of obstacles of this type is considered which are contained in a given (large) ball and have some additional properties: its connected components have bounded eccentricity, the distances between different connected components are bounded from below, and a uniform ‘no eclipse condition’ is satisfied. It is shown that if an obstacle K in this class has connected components of sufficiently small diameters, then there exists a horizontal strip near the real axis in the complex upper halfplane containing infinitely many scattering resonances (poles of the scattering matrix), i.e. the Modified LaxPhillips Conjecture holds for such K. This generalizes a wellknown result of M. Ikawa concerning balls with the same sufficiently small radius.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. An abstract meromorphicity theorem

Chapter 3. Preliminaries

Chapter 4. Ikawa’s transfer operator

Chapter 5. Resolvent estimates for transfer operators

Chapter 6. Uniform local meromorphicity

Chapter 7. Proof of the main theorem

Chapter 8. Curvature estimates


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This work deals with scattering by obstacles which are finite disjoint unions of strictly convex bodies with smooth boundaries in an odd dimensional Euclidean space. The class of obstacles of this type is considered which are contained in a given (large) ball and have some additional properties: its connected components have bounded eccentricity, the distances between different connected components are bounded from below, and a uniform ‘no eclipse condition’ is satisfied. It is shown that if an obstacle K in this class has connected components of sufficiently small diameters, then there exists a horizontal strip near the real axis in the complex upper halfplane containing infinitely many scattering resonances (poles of the scattering matrix), i.e. the Modified LaxPhillips Conjecture holds for such K. This generalizes a wellknown result of M. Ikawa concerning balls with the same sufficiently small radius.

Chapters

Chapter 1. Introduction

Chapter 2. An abstract meromorphicity theorem

Chapter 3. Preliminaries

Chapter 4. Ikawa’s transfer operator

Chapter 5. Resolvent estimates for transfer operators

Chapter 6. Uniform local meromorphicity

Chapter 7. Proof of the main theorem

Chapter 8. Curvature estimates