Advertisement

Inventiones mathematicae

, Volume 197, Issue 2, pp 417–431 | Cite as

Self-adjoint commuting ordinary differential operators

  • Andrey E. MironovEmail author
Article

Abstract

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

Mathematics Subject Classification (2010)

37K10 37K20 

Notes

Acknowledgements

The author is grateful to I.M. Krichever, O.I. Mokhov, S.P. Novikov and V.V. Sokolov for valuable discussions and stimulating interest.

References

  1. 1.
    Wallenberg, G.: Über die Vertauschbarkeit homogener linearer Differentialausdrücke. Arch. Math. Phys. 4, 252–268 (1903) zbMATHGoogle Scholar
  2. 2.
    Schur, J.: Über vertauschbare lineare Differentialausdrücke. Sitzungsber. Berl. Math. Ges. 4, 2–8 (1905) Google Scholar
  3. 3.
    Burchnall, J.L., Chaundy, I.W.: Commutative ordinary differential operators. Proc. Lond. Math. Soc. Ser. 2. 21, 420–440 (1923) CrossRefzbMATHGoogle Scholar
  4. 4.
    Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funct. Anal. Appl. 11(1), 12–26 (1977) CrossRefzbMATHGoogle Scholar
  5. 5.
    Krichever, I.M.: Commutative rings of ordinary linear differential operators. Funct. Anal. Appl. 12(3), 175–185 (1978) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Drinfeld, V.G.: Commutative subrings of certain noncommutative rings. Funct. Anal. Appl. 11(3), 9–12 (1977) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations. In: Nagata, M. (ed.) Proceedings Int. Symp. Algebraic Geometry, Kyoto, 1977, pp. 115–153. Kinokuniya Book Store, Tokyo (1978) Google Scholar
  8. 8.
    Krichever, I.M., Novikov, S.P.: Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili equation. I. Funct. Anal. Appl. 12(4), 276–286 (1978) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and nonlinear equations. Russ. Math. Surv. 35(6), 47–68 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Grinevich, P.G., Novikov, S.P.: Spectral theory of commuting operators of rank two with periodic coefficients. Funct. Anal. Appl. 16(1), 19–20 (1982) CrossRefzbMATHGoogle Scholar
  11. 11.
    Grinevich, P.G.: Rational solutions for the equation of commutation of differential operators. Funct. Anal. Appl. 16(1), 15–19 (1982) CrossRefzbMATHGoogle Scholar
  12. 12.
    Grunbaum, F.: Commuting pairs of linear ordinary differential operators of orders four and six. Physica D 31(3), 424–433 (1988) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Latham, G.: Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV. Appl. Math. Lett. 8(6), 73–78 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Latham, G., Previato, E.: Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations. Acta Appl. Math. 39, 405–433 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Mokhov, O.I.: On commutative subalgebras of Weyl algebra, which are associated with an elliptic curve. In: International Conference on Algebra in Memory of A.I. Shirshov (1921–1981), Barnaul, USSR, 20–25 August, p. 85 (1991). Reports on theory of rings, algebras and modules Google Scholar
  16. 16.
    Mokhov, O.I.: On the commutative subalgebras of Weyl algebra, which are generated by the Chebyshev polynomials. In: Third International Conference on Algebra in Memory of M.I. Kargapolov (1928–1976), Krasnoyarsk, Russia, 23–28 August 1993, p. 421. Inoprof, Krasnoyarsk (1993) Google Scholar
  17. 17.
    Previato, E., Wilson, G.: Differential operators and rank 2 bundles over elliptic curves. Compos. Math. 81(1), 107–119 (1992) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Dehornoy, P.: Opérateurs différentiels et courbes elliptiques. Compos. Math. 43(1), 71–99 (1981) zbMATHMathSciNetGoogle Scholar
  19. 19.
    Mokhov, O.I.: Commuting differential operators of rank 3 and nonlinear differential equations. Math. USSR, Izv. 35(3), 629–655 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Mironov, A.E.: A ring of commuting differential operators of rank 2 corresponding to a curve of genus 2. Sb. Math. 195(5), 711–722 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Mironov, A.E.: On commuting differential operators of rank 2. Sib. Electron. Math. Rep. 6, 533–536 (2009) zbMATHMathSciNetGoogle Scholar
  22. 22.
    Mironov, A.E.: Commuting rank 2 differential operators corresponding to a curve of genus 2. Funct. Anal. Appl. 39(3), 240–243 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Zuo, D.: Commuting differential operators of rank 3 associated to a curve of genus 2. SIGMA 8, 044 (2012) Google Scholar
  24. 24.
    Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties. Usp. Mat. Nauk 31:1(187), 55–136 (1976) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Dixmier, J.: Sur les algèbres de Weyl. Bull. Soc. Math. Fr. 96, 209–242 (1968) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Laboratory of Geometric Methods in Mathematical PhysicsMoscow State UniversityMoscowRussia

Personalised recommendations