Mathematical Notes

, Volume 99, Issue 1–2, pp 308–311 | Cite as

Common eigenfunctions of commuting differential operators of rank 2

  • V. S. OganesyanEmail author


Commuting differential operators of rank 2 are considered. With each pair of commuting operators a complex curve called the spectral curve is associated. The genus of this curve is called the genus of the commuting pair. The dimension of the space of common eigenfunctions is called the rank of the commuting operators. The case of rank 1 was studied by I. M. Krichever; there exist explicit expressions for the coefficients of commuting operators in terms of Riemann theta-functions. The case of rank 2 and genus 1 was considered and studied by S. P. Novikov and I.M. Krichever. All commuting operators of rank 3 and genus 1 were found by O. I. Mokhov. A. E. Mironov invented an effective method for constructing operators of rank 2 and genus greater than 1; by using this method, many diverse examples were constructed. Of special interest are commuting operators with polynomial coefficients, which are closely related to the Jacobian problem and many other problems. Common eigenfunctions of commuting operators with polynomial coefficients and smooth spectral curve are found explicitly in the present paper. This has not been done so far.


commuting differential operators of rank 2 common eigenfunctions spectral curve confluent Heun equation 


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  1. 1.
    J. L. Burchnall and I. W. Chaundy, “Commutative ordinary differential operators,” Proc. London Math. Soc. (2) 21, 420–440 (1923).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funktsional. Anal. Prilozhen. 11 (1), 15–31 (1977) [Functional Anal. Appl. 11 (1), 12–26 (1977)].MathSciNetzbMATHGoogle Scholar
  3. 3.
    I. M. Krichever, “Commutative rings of ordinary linear differential operators,” Funktsional. Anal. Prilozhen. 12 (3), 20–31 (1978) [Functional Anal. Appl. 12 (3), 175–185 (1978)].MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Dixmier, “Sur les algèbres deWeyl,” Bull. Soc.Math. France 96, 209–242 (1968).MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and non-linear equations,” UspekhiMat. Nauk 35 (6), 47–68 (1980) [RussianMath. Surveys 35 (6), 53–79 (1980)].MathSciNetzbMATHGoogle Scholar
  6. 6.
    O. I. Mokhov, “Commuting ordinary differential operators of rank 3 corresponding to an elliptic curve,” UspekhiMat. Nauk 37 (4), 169–170 (1982) [RussianMath. Surveys 37 (4), 129–130 (1982)].MathSciNetzbMATHGoogle Scholar
  7. 7.
    O. I. Mokhov, “Commuting differential operators of rank 3, and nonlinear differential equations,” Izv. Akad. Nauk SSSR Ser. Mat. 53 (6), 1291–1315 (1989) [Math. USSR-Izv. 35 (3), 629–655 (1990)].MathSciNetGoogle Scholar
  8. 8.
    A. E. Mironov, “Self-adjoint commuting ordinary differential operators,” Invent. Math. 197 (2), 417–431 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    O. I. Mokhov, “On commutative subalgebras of theWeyl algebra related to commuting operators of arbitrary rank and genus,” Mat. Zametki 94 (2), 314–316 (2013) [Math. Notes 94 (1–2), 298–300 (2013)].MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. S. Oganesyan, Commuting Differential Operators of Rank 2 with Polynomial Coefficients, arXiv: 1409.4058 (2015).zbMATHGoogle Scholar
  11. 11.
    S. Yu. Slavyanov and V. Lai, Special Functions. Uniform Theory Based on Analysis of Singularities (Nevskii Dialekt, St. Petersburg, 2002) [in Russian].Google Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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