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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
Article

Abstract

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.

Keywords

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.

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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

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