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The Malgrange Form and Fredholm Determinants
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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The Malgrange Form and Fredholm Determinants
Інший ID:
2010 Mathematics Subject Classification: 35Q15; 47A53; 47A68
DOI:10.3842/SIGMA.2017.046
DOI:10.3842/SIGMA.2017.046
Посилання:
The Malgrange Form and Fredholm Determinants / M. Bertola // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ.
Підтримка:
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.
Дата:
2017
Переглядів:
537
Завантажень:
220
Анотація:
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.
