Communications in Mathematical Physics

, Volume 172, Issue 2, pp 263–285 | Cite as

Classical and quantum integrable systems in 263-1263-1263-1and separation of variables

  • J. Harnad
  • P. Winternitz


Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra\(\widetilde{\mathfrak{g}\mathfrak{l}}^{ + *} (2,\mathbb{R})\) are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e. by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order\(\mathcal{O}(\hbar ^2 )\) in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. for each case — in the ambient space ℝ n , the sphere and the ellipsoid — the Schrödinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lamé type.


Neural Network Assure Phase Space Nonlinear Dynamics Integrable System 
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. [Ad] Adler, M.: On a Trace Functional for Formal Pseudo-Differential Operators and the Symplectic Structure of the Korteweg-de Vries Equation. Invent. Math.50, 219–248 (1979)Google Scholar
  2. [AHH1] Adams, M.R., Harnad, J., Hurtubise, J.: “Isospectral Hamiltonian Flows in Finite and Infinite Dimensions II. Integration of Flows.” Commun. Math. Phys.134, 555–585 (1990)Google Scholar
  3. [AHH2] Adams, M.R., Harnad, J., Hurtubise, J.: Dual Moment Maps to Loop Algebras. Lett. Math. Phys.20, 294–308 (1990)Google Scholar
  4. [AHH3] Adams, M.R., Harnad, J., Hurtubise, J.: Liouville Generating Function for Isospectral Hamiltonian Flow in Loop Algebras. In: Integrable and Superintergrable Systems, ed. B.A. Kuperschmidt, Singapore: World Scientific, 1990Google Scholar
  5. [AHH4] Adams, M.R., Harnad, J., Hurtubise, J.: Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras. Commun. Math. Phys.155, 385–413 (1993)Google Scholar
  6. [AHP] Adams, M.R., Harnad, J., Previato, E.: Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalised Moser Systems and Moment Maps into Loop Algebras. Commun. Math. Phys.117, 451–500 (1988)Google Scholar
  7. [AvM] Adler, M. and van Moerbeke, P.: Completely Integrable Systems, Euclidean Lie Algebras, and Curves. Adv. Math.38, 267–317 (1980); linearization of Hamiltonian Systems, Jacobi Varieties and Representation Theory. ibid. Adv. Math.38, 318–379 (1980)Google Scholar
  8. [BKW] Boyer, C.P., Kalnins, E.G., Winternitz, P.: Completely Integrable Relativistic Hamiltonian Systems and Separation of Variables in Hermitian Hyperbolic Spaces J. Math. Phys.24, 2022–2034 (1983)Google Scholar
  9. [BKW2] Boyer, C.P., Kalnins, E.G., Winternitz, P.: Separation of Variables for the Hamilton-Jacobi Equation on Complex Projective Spaces. SIAM J. Math. Anal.16, 93–109 (1985)Google Scholar
  10. [BT] Babelon, O., Talon, M.: Separation of variables for the classical and quantum Neumann model. Nucl. Phys.B379, 321 (1992)Google Scholar
  11. [D] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science Monograph Seris No. 2, New York, 1964Google Scholar
  12. [Du] Dubrovin, B.A.: Theta Functions and Nonlinear Eqations. Russ. Math. Surv.36, 11–92 (1981)Google Scholar
  13. [G] Gaudin, M.: Diagonalization d'-une classe d'hamiltoniens de spin. J. Physique37, 1087–1098 (1976)Google Scholar
  14. [GH] Griffiths, P., Harrris, J.: Principles of Algebraic Geometry. New York: Wiley, 1978Google Scholar
  15. [GHHW] Gagnon, L., Harnad, J., Hurtubise, J., Winternitz, P.: Abelian Integrals and the Reduction Method for an Integrable Hamiltonian System. J. Math. Phys.26, 1605–1612 (1985)Google Scholar
  16. [H] Harnad, J.: Isospectral Flow and Liouville-Arnold Integration in Loop Algebras. In: Geometric and Quantum Methods in Integrable Systems. Springer Lecture Notes in Physics 424, G. Helminck (ed.) Berlin, Heidelberg, New York: Springer, 1993Google Scholar
  17. [HW] Harnad, J., Wisse, M.-A., Isospectral Flow in Loop Algebras and Quasiperiodic Solutions to the Sine-Gordon Equation. J. Math. Phys.34, 3518–3526 (1993)Google Scholar
  18. [K] Kalnins, E.G.: Separation of Variables for Riemannian Symmetric Spaces of Constant Curvature. Pitman Monographs and Surveys in Pure and Applied Mathematics28, (1986)Google Scholar
  19. [KM] Kalnins, E.G., Miller, W. Jr.: Separation of Variables onn-dimensional Riemannian manifolds: 1. Then-sphereS n and Euclidean SpaceR n. J. Math. Phys.27, 1721–1736 (1986)Google Scholar
  20. [KMW] Kalnins, E.G., Miller, W. Jr., Winternitz, P.: The GroupO(4), Separation of Variables and the Hydrogen Atom. SIAM J. Appl. Math.30, 630–664 (1976)Google Scholar
  21. [KN] Krichever, I.M., Novikov, S.P.: Holomorphic Bundles over Algebraic Curves and Nonlinear Equations. Russ. Math. Surveys32, 53–79 (1980)Google Scholar
  22. [Ko] Kostant, G.: The Solution to a Generalized Toda Lattice and Representation. Theory. Adv. Math.34, 195–338 (1979)Google Scholar
  23. [Ku] Kuznetsov, Vadim B.: Equivalence of two graphical calculi. J. Phys. A25, 6005–6026 (1992)Google Scholar
  24. [Mc] Macfarlane, A.J.: The quantum Neumann model with the potential of Rosochatius. Nucl. Phys.B386, 453–467 (1992)Google Scholar
  25. [Mo] Moser, J.: Geometry of Quadrics and Spectral Theory, The Chern Symposium, Berkeley, June 1979, 147–188, New York: Springer, Berlin, Heidelberg 1980Google Scholar
  26. [ORW] del Olmo, M.A., Rodriguez, M.A., Winternitz, P.: Integrable Systems Based on SU(p, q) Homogeneous Manifolds. J. Math Phys.34, 5118–5139 (1993)Google Scholar
  27. [Sk1] Sklyanin, E.K.: Separation of Variables in the Glaudin Model. J. Sov. Math.47, 2473–2488 (1989)Google Scholar
  28. [Sk2] Sklyanin, E.K.: Functional Bethe Ansatz. In: Integrable and Superintergrable Systems, ed. B.A. Kupershmidt, Singapore: World Scientific, 1990Google Scholar
  29. [SK3] Sklyanin, E.K.: Separation of Variables in the Quantum Integrable Models Related to the YangianY[sl(3)]. Preprint NI-92013 (1992)Google Scholar
  30. [Sy] Symes, W.: Systems of Toda Type, Inverse Spectral Problems and Representation Theory. Invent. Math.59, 13–51 (1980)Google Scholar
  31. [TW] Tafel, J., Wisse, M.A.: Loop Algebra Approach to Generalized Sine-Gordon Equations. J. Math. Phys.34, (1993)Google Scholar
  32. [To] Toth, John A.: Various Quantum Mechanical Aspects of Quadratic Forms. M.I.T. preprint (1992)Google Scholar
  33. [W] Wisse, M.A.: Darboux Coordinates and Isospectral Hamiltonian Flows for the Massive Thirring Model. Lett. Math. Phys.28, 287–294 (1993)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • J. Harnad
    • 1
    • 2
  • P. Winternitz
    • 2
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada

Personalised recommendations