Abstract:
The problem of describing a commuting pair of differential operators in terms of its Burchnall–Chaundy curve and the holomorphic bundle over it is considered.
A characteristic of the matrix case is the occurrence of vector rank, a bundle having various dimensions over various components of the Burchnall–Chaundy curve. A complete, independent system which determines the pair of operators uniquely is chosen. In the last section, a proof is given of S. P. Novikov's criterion for an operator with periodic coefficients to be an operator of a nontrivial commuting pair.
Bibliography: 25 titles.
Citation:
P. G. Grinevich, “Vector rank of commuting matrix differential operators. Proof of S. P. Novikov's criterion”, Math. USSR-Izv., 28:3 (1987), 445–465
This publication is cited in the following 7 articles:
Emma Previato, Sonia L. Rueda, Maria-Angeles Zurro, “Burchnall–Chaundy polynomials for matrix ODOs and Picard–Vessiot Theory”, Physica D: Nonlinear Phenomena, 453 (2023), 133811
Vardan Oganesyan, “Matrix Commuting Differential Operators of Rank 2 and Arbitrary Genus”, International Mathematics Research Notices, 2019:3 (2019), 834
A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154
V. S. Oganesyan, “The AKNS hierarchy and finite-gap Schrödinger potentials”, Theoret. and Math. Phys., 196:1 (2018), 983–995
A. B. Zheglov, A. E. Mironov, “Moduli Beikera – Akhiezera, puchki Krichevera i kommutativnye koltsa differentsialnykh operatorov v chastnykh proizvodnykh”, Dalnevost. matem. zhurn., 12:1 (2012), 20–34
I. M. Krichever, S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Russian Math. Surveys, 58:3 (2003), 473–510
I. A. Taimanov, “Secants of Abelian varieties, theta functions, and soliton equations”, Russian Math. Surveys, 52:1 (1997), 147–218
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