Titlebook: Spectral Theory of Infinite-Area Hyperbolic Surfaces; David Borthwick Book 2016Latest edition Springer International Publishing Switzerlan

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https://doi.org/10.1007/978-3-319-33877-4Complex Analysis; Hyperbolic Surface; Resonance Theory; Scattering Theory; Spectral Gap; Spectral Theory;

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Selberg Theory for Finite-Area Hyperbolic Surfaces,To set the stage for the development of the spectral theory in the infinite-area case, we will first review some details of the spectral theory for compact and finite-area hyperbolic surfaces (Fuchsian groups of the first kind).

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Spectral Theory for the Hyperbolic Plane,In our discussion of spectral theory we naturally start with the hyperbolic plane itself, the primary example of a hyperbolic surface of infinite area.

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The Resolvent,In Chapters . and . we worked out the resolvent kernels for the elementary surfaces. This provides a set of model resolvents for funnels and cusps in particular, which are the only possible end types in a non-elementary geometrically finite hyperbolic surface by Theorem ..

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Spectral and Scattering Theory,The basic spectral theory of the Laplacian on a geometrically finite hyperbolic manifold was worked out by Lax-Phillips [.–.], in the abstract framework of Lax-Phillips scattering theory [.].

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