Harmonic analysis on the quantized Riemann sphere
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-...
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| Format: | Article |
| Language: | English |
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Wiley
1993-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171293000274 |
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| _version_ | 1832560407240245248 |
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| author | Jaak Peetre Genkai Zhang |
| author_facet | Jaak Peetre Genkai Zhang |
| author_sort | Jaak Peetre |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a0c2b69ec68f4d2a86d505a9fc3d1fc9 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1993-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-a0c2b69ec68f4d2a86d505a9fc3d1fc92025-02-03T01:27:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251993-01-0116222524310.1155/S0161171293000274Harmonic analysis on the quantized Riemann sphereJaak Peetre0Genkai Zhang1Matematiska institutionen, Stockholms universitet, Box 6701, Stockholm S-113 85, SwedenMatematiska institutionen, Stockholms universitet, Box 6701, Stockholm S-113 85, SwedenWe 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.http://dx.doi.org/10.1155/S0161171293000274riemann spherequantizationreproducing kernelinvariant Cauchy-Riemann operatorinvariant LaplaceanMorse operatorHankel operatorhypergeomettic functionorthogonal polynomial. |
| spellingShingle | Jaak Peetre Genkai Zhang Harmonic analysis on the quantized Riemann sphere International Journal of Mathematics and Mathematical Sciences riemann sphere quantization reproducing kernel invariant Cauchy-Riemann operator invariant Laplacean Morse operator Hankel operator hypergeomettic function orthogonal polynomial. |
| title | Harmonic analysis on the quantized Riemann sphere |
| title_full | Harmonic analysis on the quantized Riemann sphere |
| title_fullStr | Harmonic analysis on the quantized Riemann sphere |
| title_full_unstemmed | Harmonic analysis on the quantized Riemann sphere |
| title_short | Harmonic analysis on the quantized Riemann sphere |
| title_sort | harmonic analysis on the quantized riemann sphere |
| topic | riemann sphere quantization reproducing kernel invariant Cauchy-Riemann operator invariant Laplacean Morse operator Hankel operator hypergeomettic function orthogonal polynomial. |
| url | http://dx.doi.org/10.1155/S0161171293000274 |
| work_keys_str_mv | AT jaakpeetre harmonicanalysisonthequantizedriemannsphere AT genkaizhang harmonicanalysisonthequantizedriemannsphere |