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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