HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE

JAAK PEETRE ； GENKAI ZHANG

《International Journal of Mathematics and Mathematical Sciences》 1993卷 (1993/12) Pp. 225-243

https://doi.org/10.1155/S0161171293000274

摘要

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 S^2. In particular, we determine a concrete orthogonal base in the relevant Hilbert space L^(ν,2)(S^2), where -ν/2 is the degree of the form, a section of a certain holomorphic line bundle over the sphere S^2. 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.

關鍵字

Riemann sphere ； quantization ； reproducing kernel ； invariant Cauchy-Riemann operator ； invariant Laplacean ； Morse operator ； Hankel operator ； hypergeometric function ； orthogonal polynomial