Classical and quantum integrable systems in \(\widetilde{\mathfrak gl}(2)^{+*}\) and separation of variables. (English) Zbl 0842.58046

Summary: 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 \(O(\hbar^2)\) 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 \(\mathbb{R}^n\), the sphere and the ellipsoid – the Schrödinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lamé type.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q40 PDEs in connection with quantum mechanics
53D50 Geometric quantization
35Q58 Other completely integrable PDE (MSC2000)
Full Text: DOI arXiv


[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) · Zbl 0393.35058
[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) · Zbl 0717.58051
[3] [AHH2] Adams, M.R., Harnad, J., Hurtubise, J.: Dual Moment Maps to Loop Algebras. Lett. Math. Phys.20, 294–308 (1990) · Zbl 0721.58025
[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, 1990 · Zbl 0761.58022
[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) · Zbl 0791.58047
[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) · Zbl 0659.58022
[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) · Zbl 0455.58017
[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) · Zbl 0523.58023
[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) · Zbl 0569.35012
[10] [BT] Babelon, O., Talon, M.: Separation of variables for the classical and quantum Neumann model. Nucl. Phys.B379, 321 (1992)
[11] [D] Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science Monograph Seris No. 2, New York, 1964 · Zbl 0141.44603
[12] [Du] Dubrovin, B.A.: Theta Functions and Nonlinear Eqations. Russ. Math. Surv.36, 11–92 (1981) · Zbl 0549.58038
[13] [G] Gaudin, M.: Diagonalization d’-une classe d’hamiltoniens de spin. J. Physique37, 1087–1098 (1976)
[14] [GH] Griffiths, P., Harrris, J.: Principles of Algebraic Geometry. New York: Wiley, 1978
[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) · Zbl 0597.70020
[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, 1993 · Zbl 0801.35121
[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) · Zbl 0780.35100
[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) · Zbl 0658.53041
[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) · Zbl 0602.35014
[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) · Zbl 0335.35002
[21] [KN] Krichever, I.M., Novikov, S.P.: Holomorphic Bundles over Algebraic Curves and Nonlinear Equations. Russ. Math. Surveys32, 53–79 (1980) · Zbl 0548.35100
[22] [Ko] Kostant, G.: The Solution to a Generalized Toda Lattice and Representation. Theory. Adv. Math.34, 195–338 (1979) · Zbl 0433.22008
[23] [Ku] Kuznetsov, Vadim B.: Equivalence of two graphical calculi. J. Phys. A25, 6005–6026 (1992) · Zbl 0784.53056
[24] [Mc] Macfarlane, A.J.: The quantum Neumann model with the potential of Rosochatius. Nucl. Phys.B386, 453–467 (1992)
[25] [Mo] Moser, J.: Geometry of Quadrics and Spectral Theory, The Chern Symposium, Berkeley, June 1979, 147–188, New York: Springer, Berlin, Heidelberg 1980
[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) · Zbl 0783.58034
[27] [Sk1] Sklyanin, E.K.: Separation of Variables in the Glaudin Model. J. Sov. Math.47, 2473–2488 (1989) · Zbl 0692.35107
[28] [Sk2] Sklyanin, E.K.: Functional Bethe Ansatz. In: Integrable and Superintergrable Systems, ed. B.A. Kupershmidt, Singapore: World Scientific, 1990
[29] [SK3] Sklyanin, E.K.: Separation of Variables in the Quantum Integrable Models Related to the YangianY[sl(3)]. Preprint NI-92013 (1992)
[30] [Sy] Symes, W.: Systems of Toda Type, Inverse Spectral Problems and Representation Theory. Invent. Math.59, 13–51 (1980) · Zbl 0474.58009
[31] [TW] Tafel, J., Wisse, M.A.: Loop Algebra Approach to Generalized Sine-Gordon Equations. J. Math. Phys.34, (1993) · Zbl 0784.35103
[32] [To] Toth, John A.: Various Quantum Mechanical Aspects of Quadratic Forms. M.I.T. preprint (1992)
[33] [W] Wisse, M.A.: Darboux Coordinates and Isospectral Hamiltonian Flows for the Massive Thirring Model. Lett. Math. Phys.28, 287–294 (1993) · Zbl 0811.58050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.