Rilegato

ISBN 10: 0817645241 ISBN 13: 9780817645243

Casa editrice: Birkhäuser, 2007

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.

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.

Review:

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)