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 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 an introductory example that is broadly accessible.

Inhalt

Preface.- Hyperbolic surfaces.- Geometry of H.- Fuchsian groups.- Geometric finiteness.- Classification of hyperbolic ends.- Length spectrum and Selberg's zeta function.- Review of the Compact Case.- Spectral theory for compact manifolds.- Selberg's trace formula for compact surfaces.- Consequences of the trace formula.- Review of the finite-volume case.- Finite-volume hyperbolic surfaces.- Spectral theory.- Selberg's trace formula.- Scattering Theory in Model Cases.- Spectral theory of H.- Scattering theory on H.- Hyperbolic cylinders.- Funnels.- Parabolic cylinder.- Scattering Theory for infinite-volume hyperbolic surfaces.- Compactification.- Continuation of the resolvent.- Resolvent asymptotics and the stretched product.- Structure of the resolvent kernel.- Discrete and continuous spectrum.- Generalized eigenfunctions.- Scattering matrix.- Structure of kernels in the conformally compact case.- Resonances and scattering poles.- Multiplicities of resonances.- Scattering poles.- Half-integer points.- Coincidence of resonances and scattering poles.- Upper bound on the density of resonances.- Infinite-volume spectral geometry.- Relative scattering determinant.- Regularized traces.- The resolvent 0-trace calculation.- Structure of Selberg's zeta function.- The Poisson formula for resonances.- Application.- Lower bounds on the density.- Weyl formula for the scattering phase.- The length spectrum.- Finiteness of isospectral classes.- Appendix A Functional analysis.- Basic spectral theory.- Analytic Fredholm theorem.- Operator residues and multiplicities.- Appendix B Asymptotic expansions.- References.- Index.