Spectral Theory of Infinite-Area Hyperbolic Surfaces (eBook)

David Borthwick (Autor)

eBook Download: PDF

2007 | 2007
XI, 355 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4653-0 (ISBN)

Infos

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.

Inhalt

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.

Produkt-Details

• Erscheint lt. Verlag: 13.9.2007
• Reihe/Serie: Progress in Mathematics
• Reihe/Serie: Progress in Mathematics
• Zusatzinfo: XI, 355 p.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Complex Analysis • Distribution • Functional Analysis • Inverse scattering problem • Partial differential equations • resonance theory • Riemann Surfaces • scattering theory • spectral theory
• ISBN-10: 0-8176-4653-1 / 0817646531
• ISBN-13: 978-0-8176-4653-0 / 9780817646530