The Malgrange Form and Fredholm Determinants
We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function τ which is locally analytic on the space of deformations and that is expressed as a Fredholm de...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148566 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The Malgrange Form and Fredholm Determinants / M. Bertola // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862545761992966144 |
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| author | Bertola, M. |
| author_facet | Bertola, M. |
| citation_txt | The Malgrange Form and Fredholm Determinants / M. Bertola // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function τ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of ''integrable'' type in the sense of Its-Izergin-Korepin-Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle.
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| first_indexed | 2025-11-25T07:15:11Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-148566 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T07:15:11Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bertola, M. 2019-02-18T15:57:07Z 2019-02-18T15:57:07Z 2017 The Malgrange Form and Fredholm Determinants / M. Bertola // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q15; 47A53; 47A68 DOI:10.3842/SIGMA.2017.046 https://nasplib.isofts.kiev.ua/handle/123456789/148566 We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function τ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of ''integrable'' type in the sense of Its-Izergin-Korepin-Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle. The author wishes to thank Oleg Lisovyy for asking a very pertinent question on the representation of the Malgrange form in terms of Fredholm determinants. Part of the thinking was done
 during the author’s stay at the “Centro di Ricerca Matematica Ennio de Giorgi” at the Scuola
 Normale Superiore in Pisa, workshop on “Asymptotic and computational aspects of complex
 dif ferential equations” organized by G. Filipuk, D. Guzzetti and S. Michalik. The author wishes
 to thank the organizers and the Institute for providing an opportunity of fruitful exchange. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Malgrange Form and Fredholm Determinants Article published earlier |
| spellingShingle | The Malgrange Form and Fredholm Determinants Bertola, M. |
| title | The Malgrange Form and Fredholm Determinants |
| title_full | The Malgrange Form and Fredholm Determinants |
| title_fullStr | The Malgrange Form and Fredholm Determinants |
| title_full_unstemmed | The Malgrange Form and Fredholm Determinants |
| title_short | The Malgrange Form and Fredholm Determinants |
| title_sort | malgrange form and fredholm determinants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148566 |
| work_keys_str_mv | AT bertolam themalgrangeformandfredholmdeterminants AT bertolam malgrangeformandfredholmdeterminants |