Spectral Theory of Infinite-Area Hyperbolic Surfaces

David Borthwick (auth.)0817645241, 9780817645243, 9780817646530, 0817646531

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.

Table of contents :
Front Matter….Pages I-XI
Introduction….Pages 1-5
Hyperbolic Surfaces….Pages 7-35
Compact and Finite-Area Surfaces….Pages 37-48
Spectral Theory for the Hyperbolic Plane….Pages 49-59
Model Resolvents for Cylinders….Pages 61-73
TheResolvent….Pages 75-91
Spectral and Scattering Theory….Pages 93-116
Resonances and Scattering Poles….Pages 117-146
Upper Bound for Resonances….Pages 147-169
Selberg Zeta Function….Pages 171-205
Wave Trace and Poisson Formula….Pages 207-221
Resonance Asymptotics….Pages 223-235
Inverse Spectral Geometry….Pages 237-258
Patterson–Sullivan Theory….Pages 259-295
Dynamical Approach to the Zeta Function….Pages 297-314
Back Matter….Pages 315-350