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Harmonic analysis on the quantized Riemann sphere. (English) Zbl 0776.30009
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 and 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 ${L}^{\nu ,2}\left({S}^{2}\right)$, where $-\frac{\nu }{2}$ 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-plity” theory.
##### MSC:
 30C40 Kernel functions and applications (one complex variable) 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators