# Spectral Theory of Infinite-Area Hyperbolic Surfaces

Part of the Progress in Mathematics book series (PM, volume 256)

Part of the Progress in Mathematics book series (PM, volume 256)

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.

Complex analysis Distribution Riemann surfaces functional analysis inverse scattering problem resonance theory scattering theory spectral theory

- DOI https://doi.org/10.1007/978-0-8176-4653-0
- Copyright Information Birkhäuser Boston 2007
- Publisher Name Birkhäuser Boston
- eBook Packages Mathematics and Statistics
- Print ISBN 978-0-8176-4524-3
- Online ISBN 978-0-8176-4653-0
- Series Print ISSN 0743-1643
- Series Online ISSN 2296-505X
- About this book