Spectral Theory of Infinite-Area Hyperbolic Surfaces (Progress in Mathematics) - Hardcover

Borthwick, David

Hardcover

ISBN 10: 0817645241 ISBN 13: 9780817645243

Publisher: Birkhäuser, 2007

Synopsis

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.

About this title

From the Back Cover

This book introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of dramatic recent developments in the field. These developments were prompted by advances in geometric scattering theory in the early 1990s which provided new tools for the study of resonances. Hyperbolic surfaces provide an ideal context in which to introduce these new ideas, with technical difficulties kept to a minimum.

The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, spectral theory, and ergodic theory. The book highlights these connections, at a level accessible to graduate students and researchers from a wide range of fields.

Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, characterization of the spectrum, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function.

Edition details

• Publisher: Birkhäuser
• Publication date: 2007
• Language: English
• ISBN 10: 0817645241
• ISBN 13: 9780817645243
• Binding: Hardcover
• Number of pages: 355