Asymptotics of Resonances for Radial Potential Scattering in Hyperbolic Space Open Access

Crompton, Laura Catherine (2012)

Permanent URL: https://etd.library.emory.edu/concern/etds/6w924c340?locale=en
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Abstract

Abstract
Asymptotics of Resonances for Radial Potential Scattering in Hyperbolic Space
It is shown that for scattering by a spherically symmetric potential with
compact support in Hn+1 , the resonance counting function N (r) is asymp-
totic to Crn+1. C is found explicitly, and is shown to depend only on the
dimension and the radius of support of the potential.

Table of Contents

1 Introduction 1

2 Estimates on the Scattering Matrix 6

2.1 Relative Scattering Theory Background . . . . . . . . . 6

2.2 Form of the Relative Scattering Matrix . . . . . . . . . 10

2.3 Scattering Matrix Estimates . . . . . . . . . . . . . . . . 13

2.3.1 Background Estimates . . . . . . . . . . . . . . . . . . . 14

2.3.2 Estimating the Exponentially Decaying Terms . . . 17

2.3.3 Estimating the Exponentially Growing Terms . . . . 20

2.3.4 Estimating the Leading Exponential Growth Term . 23

2.4 Proof of the Bound on the Scattering Matrix . . . . . 27

2.4.1 Estimating the Scattering Matrix for Small k . . . . 29

3 Asymptotics for the Number of Scattering Poles . . . . 31

Appendix 40

4.1 Scattering Determinant Estimate . . . . . . . . . . . . 40

4.2 Scattering Phase Estimate . . . . . . . . . . . . . . . . . 40

4.3 Laplace's Method . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 48

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