Harmonics on Hyperbolic space

Question

I would like to know if there exists an analogue for hyperbolic space of the so called spherical harmonics which play a major role in the quantum states construction in a hydrogen atom. In other words are there 'hyperbolic harmonics' and how trivial is it to obtain them?

Answer 1

As I understood the question, you would like to find a solutions $u$ of the hyperbolic laplacian $\Delta_{h} u=0$ that are harmonic homogenous polynomials such that every function that is solution of the hyperbolic laplacian can be represented as a sum of these hyperbolic harmonic homogenous polynomials.

Every hyperbolic harmonic function can be represented as some combination of ordinary spherical harmonics, but what about hyperbolic spherical harmonics, that are solutions of the hyperbolic laplacian and also homogenous polynomials.

For further reading try

• 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

Answer 2

The question as it is phrased seems to be about the angular part, and has a simple answer: the spherical harmonics are trivially the same. On the contrary, the radial part is highly nontrivial. The basic problem was apparently first solved by Selberg. And the most sophisticated formulae for the radial part can be found in the paper by J.D.Fay, from 1976.