International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 225-243
doi:10.1155/S0161171293000274
Harmonic analysis on the quantized Riemann sphere
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.
Abstract
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.