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.

Zusammenfassung

This book introduces geometric spectral theory in the context of Riemann surfaces. A comprehensive account is given of dramatic recent developments in the context of infinite-area hyperbolic surfaces. 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.

Inhalt
Hyperbolic Surfaces.- Compact and Finite-Area Surfaces.- Spectral Theory for the Hyperbolic Plane.- Model Resolvents for Cylinders.- TheResolvent.- Spectral and Scattering Theory.- Resonances and Scattering Poles.- Upper Bound for Resonances.- Selberg Zeta Function.- Wave Trace and Poisson Formula.- Resonance Asymptotics.- Inverse Spectral Geometry.- PattersonSullivan Theory.- Dynamical Approach to the Zeta Function.