Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends Open Access

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

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	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

Last modified

Primary PDF

Thumbnail	Title	Date Uploaded	Actions
	Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends ()	2018-08-28 11:39:33 -0400	Press to Select an action Download

Supplemental Files

Thumbnail	Title	Date Uploaded	Actions