Theoretical and Mathematical Physics

, Volume 140, Issue 3, pp 1283–1298 | Cite as

Polynomial Conservation Laws in Quantum Systems

  • V. V. Kozlov
  • D. V. TreshchevEmail author


We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e., the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.

Hamiltonian operator polynomial differential operator system with exponential interaction potential spectrum 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. V. Kozlov, Thermal Equilibrium According to Gibbs and Poincaré[in Russian], Institute of Computer Studies, Moscow-Izhevsk (2002).Google Scholar
  2. 2.
    V. V. Kozlov and D. V. Treshchev, Math. USSR Sb., 63, 121–139 (1989).Google Scholar
  3. 3.
    V. V. Kozlov, Russ. Acad. Sci., Dokl., Math., 62, 129–130 (2000).Google Scholar
  4. 4.
    M. Adler and P. van Moerbeke, Adv. Math., 38, 267–317 (1980); O. I. Bogoyavlensky, Comm. Math. Phys., 51, 201-209 (1976).CrossRefGoogle Scholar
  5. 5.
    M. Adler and P. van Moerbeke, Comm. Math. Phys., 83, 83–106 (1982); M. A. Olshanetsky and A. M. Perelomov, Invent. Math., 37, 93-108 (1976); 54, 261-269 (1979).Google Scholar
  6. 6.
    V. V. Kozlov and D. V. Treshchev, Math. USSR, Izv., 34, 555–574 (1990).Google Scholar
  7. 7.
    S. Gravel, Theor. Math. Phys., 137, 1439–1447 (2003).CrossRefGoogle Scholar
  8. 8.
    M. A. Olshanetskii, A. M. Perelomov, A. G. Reiman, and M. A. Semenov-Tyan-Shanskii, “Integrable systems: II,” in: Itogi Nauki i Tekhniki: Current Problems in Mathematics: Fundamental Directions[in Russian], Vol. 16, VINITI, Moscow (1987), p. 86–226; English transl.: M. A. Olshanetskii and A. M. Perelomov, “Integrable systems and infinite dimensional Lie algebras,” in: Dynamical System VII: Integrable Systems. Nonholomorphic Dynamical Systems(Encycl. Math. Sci., Vol. 16, V. I. Arnol’d and S. P. Novikov, eds.), Springer, Berlin (1994), pp. 87-116; A. G. Reiman and M. A. Semenov-Tyan-Shanskii, “Group-theoretical methods in the theory of finite-dimensional integrable systems,” ibid., pp. 116-225; M. A. Semenov-Tyan-Shanskii, “Quantization of open Toda lattices,” ibid., pp. 226-259.Google Scholar
  9. 9.
    B. Kostant, “Quantization and representation theory,” in: Representation Theory of Lie Groups (London Math. Soc. Lect. Notes, Vol. 34, M. F. Atiyah and G. L. Luke, eds.), Cambridge Univ. Press, New York (1979), p. 287–316; M. A. Semenov-Tyan-Shanskii, “Quantum Toda chains: Decomposition theorems and scattering [in Russian],” LOMI Preprint (1984); R. Goodman and N. R. Wallach, Comm. Math. Phys., 83, 355-386 (1982); 94, 177-217 (1984).Google Scholar
  10. 10.
    P. K. Rashevsky, Uchen. Zapiski Mosk. Ped. Inst. im. K. Libknekhta, Ser. fiz.-matem. nauk, No. 2, 83–94 (1938).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations