Hyperfunctions in hyperbolic geometry

Abstract

In the framework of real hyperbolic geometry, this review note begins with the Helgason correspondence induced by the Poisson transform between eigenfunctions of the Laplace–Beltrami operator on the hyperbolic space and hyperfunctions on its boundary at infinity . Focused on the scattering operator for real hyperbolic manifolds of finite geometry, discussion is given on the two different constructions (pseudo-differential calculus for degenerate operators and harmonic analysis for the conformal group) and some applications (Selberg zeta functions, resonances and scattering poles).

Keywords:

• hyperbolic geometry
• scattering operator
• hyperfunctions
• resonances
• Selberg’s zeta function

AMS Subject Classifications:

• 58J50
• 35P25
• 47A40

Acknowledgments

This note is a written exposition of a talk given at the conference organized on December 14–16, 2011 in honor of Professor Akira Kaneko, who has been my first teacher in graduate studies. I am grateful to the organizers for their invitation.