Spectral Theory of Infinite-area Hyperbolic Surfaces

The second edition of this text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. The discussion is limited to surfaces with hyperbolic ends as they provide an ideal context for study, keeping technical difficulties to a minimum. All of the material from the first edition is included and has been updated where appropriate, and several new sections have been added on the latest developments in the field.

Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, 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. New chapters on Naud’s proof of a spectral gap for convex co-compact hyperbolic surfaces and on dispersive phenomena and wave decay in the context of hyperbolic space have been added, as well as more up-to-date results on resonance asymptotics near the critical line and sharp geometric constants. The author also makes use of new techniques for resonance plotting that allow for resonance plots which better illuminate the existing results and conjectures on resonance distribution.

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, and ergodic theory. As such, it will be a valuable resource for graduate students and researchers from these and other related fields. 

Review of the first edition:

"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)