What's a good reference for the simplest case? I'm interested in the spectral theory of the Laplace-Beltrami operator on the upper half plane (domain, self-adjoint extension, etc.). I only need this for the 2D case, not the quotient space with a group (or other more elaborate cases).
2026-03-31 23:51:17.1775001077
Study of the Laplacian on the Hyperbolic plane
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Here are a few references for the study of the hyperbolic Laplacian. I'm not sure that any of them are exactly what you're looking for, but they should paint the picture well enough.
Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications, vol. I and II
Peter Buser, Geometry and Spectra of Compact Riemann Surfaces
Isaac Chavel, Eigenvalues in Riemannian Geometry
Edited much later: Another very good reference that I've recently come across is
Daniel Bump, Automorphic Forms and Representations
Gelfand, Graev, and Piatetsky-Shapiro, Representation Theory and Automorphic Functions, may also be of interest to you.