Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices
We revisit the symplectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szegő kernel on the spectral curve. Using variational formulas for the Szegő kernel, we construct a new system of action-angl...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212027 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices. Marco Bertola, Dmitry Korotkin and Ramtin Sasani. SIGMA 19 (2023), 104, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862577035540430848 |
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| author | Bertola, Marco Korotkin, Dmitry Sasani, Ramtin |
| author_facet | Bertola, Marco Korotkin, Dmitry Sasani, Ramtin |
| citation_txt | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices. Marco Bertola, Dmitry Korotkin and Ramtin Sasani. SIGMA 19 (2023), 104, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We revisit the symplectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szegő kernel on the spectral curve. Using variational formulas for the Szegő kernel, we construct a new system of action-angle variables for the canonical symplectic form on the space of such functions. Comparison with previously known action-angle variables shows that the vector of Riemann constants is the gradient of some function on the moduli space of spectral curves; this function is found in the case of matrix dimension 2, when the spectral curve is hyperelliptic.
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| first_indexed | 2026-03-13T15:15:19Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212027 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T15:15:19Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bertola, Marco Korotkin, Dmitry Sasani, Ramtin 2026-01-23T10:07:52Z 2023 Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices. Marco Bertola, Dmitry Korotkin and Ramtin Sasani. SIGMA 19 (2023), 104, 22 pages 1815-0659 2020 Mathematics Subject Classification: 53D30; 34M45 arXiv:2303.05602 https://nasplib.isofts.kiev.ua/handle/123456789/212027 https://doi.org/10.3842/SIGMA.2023.104 We revisit the symplectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szegő kernel on the spectral curve. Using variational formulas for the Szegő kernel, we construct a new system of action-angle variables for the canonical symplectic form on the space of such functions. Comparison with previously known action-angle variables shows that the vector of Riemann constants is the gradient of some function on the moduli space of spectral curves; this function is found in the case of matrix dimension 2, when the spectral curve is hyperelliptic. The work of M.B. was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2016-06660. The work of D.K. was supported in part by the NSERC grant RGPIN-2020-06816. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices Article published earlier |
| spellingShingle | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices Bertola, Marco Korotkin, Dmitry Sasani, Ramtin |
| title | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices |
| title_full | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices |
| title_fullStr | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices |
| title_full_unstemmed | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices |
| title_short | Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices |
| title_sort | szegő kernel and symplectic aspects of spectral transform for extended spaces of rational matrices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212027 |
| work_keys_str_mv | AT bertolamarco szegokernelandsymplecticaspectsofspectraltransformforextendedspacesofrationalmatrices AT korotkindmitry szegokernelandsymplecticaspectsofspectraltransformforextendedspacesofrationalmatrices AT sasaniramtin szegokernelandsymplecticaspectsofspectraltransformforextendedspacesofrationalmatrices |