Asymptotics of Resonances for Radial Potential Scattering in Hyperbolic Space Öffentlichkeit
Crompton, Laura Catherine (2012)
Published
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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