Skip to main content
SpringerLink
Log in

Commutative rings of partial differential operators and Lie algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We give examples of finite gap Schrödinger operators in the two-dimensional case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Burchnal, J.L., Chaundy, T.W.: Commutative ordinary differential operators. I, II. Proc. Lond. Math. Soc.21, 420–440 (1922); Proc. Royal Soc. Lond.118, 557–583 (1928)

    Google Scholar 

  2. Novikov, S.P.: Periodic problem for Korteweg-de Vries equation. I. Funct. Anal. i ego Pril.8, 54–66 (1974)

    Article  Google Scholar 

  3. Lax, P.: Periodic solution of the KdV equation. Commun. Pure Appl. Math.28, 141–188 (1975)

    Google Scholar 

  4. Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators and abelian varieties. Russ. Math. Surv.31, 51–125 (1976)

    Google Scholar 

  5. Krichever, I.M.: Commutative rings of ordinary linear differential operators. Funct. Anal. i ego Pril.12, 12–31 (1978)

    Google Scholar 

  6. Grinevich, P.G.: Ph. D. Thesis, Moscow Univ., 1985

  7. Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Schrödinger equation in magnetic field and Riemann surfaces. Sov. Math. Dokl.17, 947–951 (1976)

    Google Scholar 

  8. Novikov, S.P., Veselov, A.P.: Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Physica18D, 267–273 (1986)

    Google Scholar 

  9. Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Usp. Mat. Nauk32, 198 (1977). English transl. in Russ. Math. Surv.32, •• (1977)

    Google Scholar 

  10. Bourbaki, N.: Groupes et algebres de Lie, Chaps. 4–6. Paris: Hermann 1968

    Google Scholar 

  11. Calogero, F.: Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys.12, 419–436 (1971)

    Article  Google Scholar 

  12. Sutherland, B.: Exact results for a quantum many-body problem in one-dimension. Phys. Rev. A4, 2019–2021 (1971); Phys. Rev.A5, 1372–1376 (1972)

    Article  Google Scholar 

  13. Moser, J.: Three integrable Hamiltonian systems, connected with isospectral deformations. Adv. Math.16, 1–23 (1975)

    Article  Google Scholar 

  14. Calogero, F.: Exactly solvable one-dimensional many-body problems. Lett. Nuovo Cimento13, 411–416 (1975)

    Google Scholar 

  15. Olshanetsky, M.A., Perelomov A.M.: Completely integrable Hamiltonian systems associated with semisimple Lie algebras. Inv. Math.37, 93–108 (1976)

    Article  Google Scholar 

  16. Olshanetsky, M.A., Perelomov, A.M.: Quantum completely integrable systems connected with semisimple Lie algebras. Lett. Math. Phys.2, 7–13 (1977)

    Article  Google Scholar 

  17. Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep.94, 313–404 (1983)

    Article  Google Scholar 

  18. Veselov, A.P.: Integrable polynomial mappings and Lie algebras. In the book: “Geometry, differential equations and mechanics.” Izdat. Moscow Univ., Moscow, 1986 (Russian)

    Google Scholar 

  19. Veselov, A.P.: Integrable mappings and Lie algebras. Sov. Math. Dokl.35, 211–213 (1987)

    Google Scholar 

  20. Julia, G.: Memoire sur la permutabilité des fraction rationelles. Ann. Sci. Ecole Norm. Sup.39, 131–215 (1922)

    MathSciNet  Google Scholar 

  21. Fatou, P.: Sur l'iteration analytique et les substaitution permutables. J. Math. Pure Appl.3, 1–49 (1924)

    Google Scholar 

  22. Ritt, J.: Permutable rational functions. Trans. Am. Math. Soc.25, 339–448 (1923)

    Google Scholar 

  23. Veselov, A.P.: The dynamics of mappings of toric varieties connected with Lie algebras. Proc. of Top. Conf. (Baku 1987)

  24. Chalykh, O.A.: On some properties of polynomial mappings connected with Lie Algebras. Vestnik Mosc. Univ.3, 57–59 (1988)

    Google Scholar 

  25. Looijenga, E.: Root systems and elliptic curves. Invent. Math.38, 17–32 (1976)

    Article  Google Scholar 

  26. Bernstein, I.N., Schwarzman, O.V.: Chevalley theorem for the complex crystallographical Coxeter groups. Funct. Anal. i Prilozen12, 79–80 (1978)

    Google Scholar 

  27. Bogoyavlensky, O.I.: On perturbations of the Toda lattice. Commun. Math. Phys.51, 201–209 (1976)

    Article  Google Scholar 

  28. Veselov, A.P., Chalykh, O.A.: Quantum Calogero problem and commutative rings of many-dimensional differential operators. Usp. Mat. Nauk43, 173 (1988) (Russian)

    Google Scholar 

  29. Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions. I. Comp. Math.64, 329–352 (1987)

    Google Scholar 

  30. Heckman, G.J.: Root systems and hypergeometric functions. II. Comp. Math.64, 353–373 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, SU 117 234, Moscow, USSR

    O. A. Chalykh & A. P. Veselov

Authors
  1. O. A. Chalykh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. P. Veselov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chalykh, O.A., Veselov, A.P. Commutative rings of partial differential operators and Lie algebras. Commun.Math. Phys. 126, 597–611 (1990). https://doi.org/10.1007/BF02125702

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02125702

Keywords

Search

Navigation