International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 225-243

Harmonic analysis on the quantized Riemann sphere

Jaak Peetre and Genkai Zhang

Matematiska institutionen, Stockholms universitet, Box 6701, Stockholm S-113 85, Sweden

Received 12 February 1992

Copyright © 1993 Jaak Peetre and Genkai Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We extend the spectral analysis of differential forms on the disk (viewed as the non-Euclidean plane) in recent work by J. Peetre L. Peng G. Zhang to the dual situation of the Riemann sphere S2. In particular, we determine a concrete orthogonal base in the relevant Hilbert space Lν,2(S2), where ν2-is the degree of the form, a section of a certain holomorphic line bundle over the sphere S2. It turns out that the eigenvalue problem of the corresponding invariant Laplacean is equivalent to an infinite system of one dimensional Schrödinger operators. They correspond to the Morse potential in the case of the disk. In the course of the discussion many special functions (hypergeometric functions, orthogonal polynomials etc.) come up. We give also an application to “Ha-plitz” theory.