Spectral Theory of Infinite-Area Hyperbolic Surfaces

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This text introduces geometric spectral theory within infinite-area Riemann surfaces, offering a detailed overview of recent advancements in the field. The second edition expands the context to general surfaces with hyperbolic ends, creating a conducive environment for the development of spectral theory while minimizing technical challenges. It includes all updated material from the first edition, along with new sections. Key topics encompass the geometry of hyperbolic surfaces, resolvent analysis of the Laplacian, scattering theory, resonances, the Selberg zeta function, the Poisson formula, resonance distribution, the inverse scattering problem, Patterson-Sullivan theory, and a dynamical approach to the zeta function. New sections highlight the latest developments, such as the spectral gap, resonance asymptotics near the critical line, and geometric constants for resonance bounds. A new chapter presents innovative techniques for resonance calculation, shedding light on existing results and conjectures regarding resonance distribution. The spectral theory of hyperbolic surfaces intersects various disciplines, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book serves as an essential resource for graduate students and researchers in these and related fields. The first edition was praised for its clarity, thoroughness, and accessibility to a wide rang