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.
Rezensionen / Stimmen
From the reviews: "The core of the book under review is devoted to the detailed description of the Guillope-Zworski papers ... . The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed ... . The book gathers together some material which is not always easily available in the literature ... . To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader ... would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Research
Produkt-Hinweis
Illustrationen
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 20 mm
Gewicht
ISBN-13
978-0-8176-4524-3 (9780817645243)
DOI
10.1007/978-0-8176-4653-0
Schweitzer Klassifikation
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.
