# Spectral Theory of Infinite-Area Hyperbolic Surfaces [electronic resource] / by David Borthwick.

##### By: Borthwick, David [author.]

##### Contributor(s): SpringerLink (Online service)

Material type: TextSeries: Progress in Mathematics: 256Publisher: Boston, MA : Birkhäuser Boston, 2007Description: XI, 355 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817646530Subject(s): Mathematics | Functional analysis | Functions of complex variables | Partial differential equations | Differential geometry | Physics | Mathematics | Functional Analysis | Partial Differential Equations | Functions of a Complex Variable | Differential Geometry | Mathematical Methods in PhysicsAdditional physical formats: Printed edition:: No titleDDC classification: 515.7 LOC classification: QA319-329.9Online resources: Click here to access onlineItem type | Current location | Collection | Call number | Status | Date due | Barcode | Item holds |
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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 -- Patterson–Sullivan Theory -- Dynamical Approach to the Zeta Function.

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.

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