This textual content introduces geometric spectral thought within the context of infinite-area Riemann surfaces, delivering a complete account of the newest advancements within the box. For the second one variation the context has been prolonged to normal surfaces with hyperbolic ends, which gives a common surroundings for improvement of the spectral idea whereas nonetheless preserving technical problems to a minimal. All of the fabric from the 1st variation is incorporated and up to date, and new sections were added.

Topics coated comprise an creation to the geometry of hyperbolic surfaces, research of the resolvent of the Laplacian, scattering thought, resonances and scattering poles, the Selberg zeta functionality, the Poisson formulation, distribution of resonances, the inverse scattering challenge, Patterson-Sullivan concept, and the dynamical method of the zeta functionality. The new sections conceal the most recent advancements within the box, together with the spectral hole, resonance asymptotics close to the severe line, and sharp geometric constants for resonance bounds. A new bankruptcy introduces lately built options for resonance calculation that remove darkness from the prevailing effects and conjectures on resonance distribution.

The spectral thought of hyperbolic surfaces is some extent of intersection for an exceptional number of parts, together with quantum physics, discrete teams, differential geometry, quantity idea, complicated research, and ergodic idea. This ebook will function a worthwhile source for graduate scholars and researchers from those and different comparable fields.

Review of the 1st edition:

"The exposition is especially transparent and thorough, and basically self-contained; the proofs are detailed...The publication gathers jointly a few fabric which isn't constantly simply on hand within the literature...To finish, the ebook is definitely at a degree obtainable to graduate scholars and researchers from a slightly huge variety of fields. basically, the reader...would definitely profit significantly from it." (Colin Guillarmou, Mathematical reports, factor 2008 h)