Spectral Theory of Infinite-Area Hyperbolic Surfaces

1. Auflage 2007
ISBN: 978-0-8176-4524-3
Verlag: Birkhäuser

Buch, Englisch, Band 256, 355 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 713 g

Reihe: Progress in Mathematics

This book is a self-contained monograph on spectral theory for non-compact Riemann surfaces, focused on the infinite-volume case. By focusing on the scattering theory of hyperbolic surfaces, this work provides a compelling introductory example which will be accessible to a broad audience. The book opens with an introduction to the geometry of hyperbolic surfaces. Then a thorough development of the spectral theory of a geometrically finite hyperbolic surface of infinite volume is given. The final sections include recent developments for which no thorough account exists.

Zielgruppe

Research

Autoren/Hrsg.

Borthwick, David

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Naturwissenschaften Physik Quantenphysik
• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

Preface.- Hyperbolic surfaces.- Geometry of H.- Fuchsian groups.- Geometric finiteness.- Classification of hyperbolic ends.- Length spectrum and Selberg’s zeta function.- Review of the Compact Case.- Spectral theory for compact manifolds.- Selberg’s trace formula for compact surfaces.- Consequences of the trace formula.- Review of the finite-volume case.- Finite-volume hyperbolic surfaces.- Spectral theory.- Selberg’s trace formula.- Scattering Theory in Model Cases.- Spectral theory of H.- Scattering theory on H.- Hyperbolic cylinders.- Funnels.- Parabolic cylinder.- Scattering Theory for infinite-volume hyperbolic surfaces.- Compactification.- Continuation of the resolvent.- Resolvent asymptotics and the stretched product.- Structure of the resolvent kernel.- Discrete and continuous spectrum.- Generalized eigenfunctions.- Scattering matrix.- Structure of kernels in the conformally compact case.- Resonances and scattering poles.- Multiplicities of resonances.- Scattering poles.- Half-integer points.- Coincidence of resonances and scattering poles.- Upper bound on the density of resonances.- Infinite-volume spectral geometry.- Relative scattering determinant.- Regularized traces.- The resolvent 0-trace calculation.- Structure of Selberg’s zeta function.- The Poisson formula for resonances.- Application.- Lower bounds on the density.- Weyl formula for the scattering phase.- The length spectrum.- Finiteness of isospectral classes.- Appendix A Functional analysis.- Basic spectral theory.- Analytic Fredholm theorem.- Operator residues and multiplicities.- Appendix B Asymptotic expansions.- References.- Index.