Spectral Theory of Infinite-Area Hyperbolic Surfaces: 256 - Rilegato

Borthwick, David

Rilegato

ISBN 10: 0817645241 ISBN 13: 9780817645243

Casa editrice: Birkhauser, 2007

Riassunto

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.

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Recensione:

From the reviews:

"The core of the book under review is devoted to the detailed description of the Guillopé-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)

Contenuti:

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.

Su questa edizione

• Editore: Birkhauser
• Data di pubblicazione: 2007
• ISBN 10: 0817645241
• ISBN 13: 9780817645243
• Rilegatura: Copertina rigida
• Numero edizione: 1
• Numero di pagine: 355