# Scattering Resonances for Several Small Convex Bodies and the Lax-Phillips Conjecture

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*Luchezar Stoyanov*

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 half-plane containing infinitely many scattering resonances (poles of the scattering matrix), i.e. the Modified Lax-Phillips Conjecture holds for such K. This generalizes a well-known result of M. Ikawa concerning balls with the same sufficiently small radius.

#### Table of Contents

# Table of Contents

## Scattering Resonances for Several Small Convex Bodies and the Lax-Phillips Conjecture

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. An abstract meromorphicity theorem 714 free
- Chapter 3. Preliminaries 916
- Chapter 4. Ikawa's transfer operator 1522
- Chapter 5. Resolvent estimates for transfer operators 2734
- Chapter 6. Uniform local meromorphicity 3542
- Chapter 7. Proof of the Main Theorem 4754
- Chapter 8. Curvature estimates 5966
- Bibliography 7582