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

This article is cited in 23 scientific papers (total in 23 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.


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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
\by I.~M.~Krichever, S.~P.~Novikov
\paper Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles
\jour Uspekhi Mat. Nauk
\yr 2003
\vol 58
\issue 3(351)
\pages 51--88
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 3
\pages 473--510

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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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    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
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    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
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    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  elib
    22. G. S. Mauleshova, A. E. Mironov, “Difference Krichever–Novikov Operators of Rank 2”, Proc. Steklov Inst. Math., 305 (2019), 195–208  mathnet  crossref  crossref  mathscinet  isi  elib
    23. Gulnara S. Mauleshova, Andrey E. Mironov, “Discretization of Commuting Ordinary Differential Operators of Rank 2 in the Case of Elliptic Spectral Curves”, Proc. Steklov Inst. Math., 310 (2020), 202–213  mathnet  crossref  crossref  isi  elib
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