Doklady Mathematics

, Volume 93, Issue 1, pp 62–64 | Cite as

One-point commuting difference operators of rank 1

  • G. S. MauleshovaEmail author
  • A. E. Mironov


One-point commuting difference operators of rank 1 are considered. The coefficients in such operators depend on one functional parameter, and the degrees of shift operators in difference operators are positive. These operators are studied in the case of hyperelliptic spectral curves, where the base point coincides with a point of branching. Examples of operators with polynomial and trigonometric coefficients are constructed. Operators with polynomial coefficients are embedded in differential operators with polynomial coefficients. This construction provides a new method for constructing commutative subalgebras in the first Weyl algebra.


Base Point Difference Operator Point Operator Spectral Curve Shift Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. M. Krichever, Russ. Math. Surveys 33 4, 255–256 (1978).MathSciNetCrossRefGoogle Scholar
  2. 2.
    I. M. Krichever and S. P. Novikov, Russ. Math. Surveys 58 3, 473–510 (2003).MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Mumford, in Proceeding of International Symposium on Algebraic Geometry, Kyoto, Japan, 1977 (Kinokuniya, Tokio, 1978), pp. 115–153.Google Scholar
  4. 4.
    G. S. Mauleshova and A. E. Mironov, Russ. Math. Surveys 70 3, 557–559 (2015).MathSciNetCrossRefGoogle Scholar
  5. 5.
    I. M. Krichever, Dokl. Akad. Nauk SSSR 285 1, 31–36 (1985).MathSciNetGoogle Scholar
  6. 6.
    I. M. Krichever, Funct. Anal. Appl. 49 3, 175–188 (2015).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Dixmier, Bull. Soc. Math. France 96, 209–242 (1968).MathSciNetGoogle Scholar
  8. 8.
    A. Ya. Kanel-Belov and M. L. Kontsevich, Mosc. Math. J. 7 2, 209–218 (2007).MathSciNetGoogle Scholar
  9. 9.
    A. E. Mironov and A. B. Zheglov, IMRN (2015); doi:10.1093/imrn/rnv218.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia
  2. 2.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

Personalised recommendations