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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 3(351), Pages 51–88 (Mi umn628)  

This article is cited in 21 scientific papers (total in 21 papers)

Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles

I. M. Kricheverabc, S. P. Novikovbd

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
c Columbia University
d University of Maryland

Abstract: Higher-rank solutions of the equations of the two-dimensionalized Toda lattice are constructed. The construction of these solutions is based on the theory of commuting difference operators, which is developed in the first part of the paper. It is shown that the problem of recovering the coefficients of commuting operators can be effectively solved by means of the equations of the discrete dynamics of the Tyurin parameters characterizing the stable holomorphic vector bundles over an algebraic curve.

DOI: https://doi.org/10.4213/rm628

Full text: PDF file (442 kB)
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English version:
Russian Mathematical Surveys, 2003, 58:3, 473–510

Bibliographic databases:

UDC: 517.9
MSC: Primary 37K10, 47B39; Secondary 14H70, 37K20, 14H52, 14C40, 37K40, 35Q53, 81R12
Received: 15.04.2003

Citation: I. M. Krichever, S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Uspekhi Mat. Nauk, 58:3(351) (2003), 51–88; Russian Math. Surveys, 58:3 (2003), 473–510

Citation in format AMSBIB
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    2. A. E. Mironov, “Commuting difference operators with polynomial coefficients”, Russian Math. Surveys, 62:4 (2007), 819–820  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. E. Mironov, “Discrete analogues of Dixmier operators”, Sb. Math., 198:10 (2007), 1433–1442  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. Bertola M., Gekhtman M., “Effective inverse spectral problem for rational Lax matrices and applications”, Int. Math. Res. Not. IMRN, 2007, no. 23, rnm103, 39 pp.  crossref  mathscinet  zmath  isi
    5. O. K. Sheinman, “Krichever–Novikov Algebras, their Representations and Applications in Geometry and Mathematical Physics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S85–S161  mathnet  crossref  crossref  zmath
    6. Schlichenmaier M., “Higher Genus Affine Lie Algebras of Krichever - Novikov Type”, Difference Equations, Special Functions and Orthogonal Polynomials, 2007, 589–599  crossref  mathscinet  zmath  isi
    7. Matveev V.B., “30 years of finite-gap integration theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 837–875  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Bertola M., Mo M.Y., “Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights”, Adv. Math., 220:1 (2009), 154–218  crossref  mathscinet  zmath  isi  elib
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    11. A. E. Mironov, A. Nakayashiki, “Discretization of Baker–Akhiezer modules and commuting difference operators in several discrete variables”, Trans. Moscow Math. Soc., 74 (2013), 261–279  mathnet  crossref  mathscinet  zmath  elib
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    13. V. N. Davletshina, “Self-Adjoint Commuting Differential Operators of Rank 2 and Their Deformations Given by Soliton Equations”, Math. Notes, 97:3 (2015), 333–340  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. G. S. Mauleshova, A. E. Mironov, “Commuting difference operators of rank two”, Russian Math. Surveys, 70:3 (2015), 557–559  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. I. M. Krichever, “Commuting Difference Operators and the Combinatorial Gale Transform”, Funct. Anal. Appl., 49:3 (2015), 175–188  mathnet  crossref  crossref  isi  elib
    16. Morier-Genoud S., Ovsienko V., Tabachnikov S., “Introducing Supersymmetric Frieze Patterns and Linear Difference Operators”, 281, no. 3-4, 2015, 1061–1087  crossref  mathscinet  zmath  isi
    17. A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russian Math. Surveys, 71:4 (2016), 751–779  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    18. Mauleshova G.S. Mironov A.E., “One-point commuting difference operators of rank 1”, Dokl. Math., 93:1 (2016), 62–64  crossref  mathscinet  zmath  isi  elib  scopus
    19. Mauleshova G.S. Mironov A.E., “One-Point Commuting Difference Operators of Rank One and Their Relation With Finite-Gap Schrodinger Operators”, Dokl. Math., 97:1 (2018), 62–64  crossref  zmath  isi  scopus
    20. A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154  mathnet  crossref  crossref  adsnasa  isi  elib
    21. G. S. Mauleshova, “The dressing chain and one-point commuting difference operators of rank 1”, Siberian Math. J., 59:5 (2018), 901–908  mathnet  crossref  crossref  isi
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