Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends Público

Abstract

We study the spectral theory of asymptotically hyperbolic manifolds with
ends of warped-product type. Our main result is an upper bound on the
resonance counting function, with a geometric constant expressed in terms
of the respective Weyl constants for the core of the manifold and the base
manifold defining the ends. As part of this analysis, we derive asymptotic
expansions of the modified Bessel functions of complex order.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The model case . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Poisson operator . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Bessel function estimates . . . . . . . . . . . . . . . . . . . 14
3.1 Applications of Proposition 3.1 . . . . . . . . . . . . . . 16
3.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . 21
4 Resonance order of growth. . . . . . . . . . . . . . . . . . . 27
4.1 Spectral operator estimates . . . . . . . . . . . . . . . . 27
4.2 Resonance counting estimate . . . . . . . . . . . . . . . 33
5 Poisson formula . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 Sharp upper bounds . . . . . . . . . . . . . . . . . . . . . . . 43
6.1 Asymptotic counting for the model space . . . . . . . 43
6.2 Estimate of the scattering determinant . . . . . . . . . 51
6.3 Completing the sharp estimate . . . . . . . . . . . . . . 55
A Asymptotics of the sectional curvatures . . . . . . . . . 56
B Asymptotic behavior of the Airy function . . . . . . . . 58
C The method of successive approximations . . . . . . . 60
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palabra Clave	spectral theory asymptotically hyperbolic manifolds Bessel functions
Committee Chair / Thesis Advisor	Borthwick, David, Emory University
Committee Members	Benzi, Michele, Emory University Oliker, Vladimir, Emory University

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