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Toward a classification of semidegenerate 3D superintegrable systems. (English) Zbl 1361.81065


MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
17A45 Quadratic algebras (but not quadratic Jordan algebras)
17B81 Applications of Lie (super)algebras to physics, etc.
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References:

[1] Tempesta P, Winternitz P, Miller W and Pogosyan G (ed) 2005 Superintegrability in Classical and Quantum Systems vol 37 (Providence, RI: American Mathematical Society)
[2] Miller W Jr, Post S and Winternitz P 2013 Classical and quantum superintegrability with applications J. Phys. A: Math. Theor.46 423001 · Zbl 1276.81070
[3] Kalnins E G, Kress J M and Miller W Jr 2005 Second order superintegrable systems in conformally flat spaces. I: 2D classical structure theory J. Math. Phys.46 053509 · Zbl 1110.37054
[4] Kalnins E G, Kress J M and Miller W Jr 2005 II: The classical 2D Stäckel transform J. Math. Phys.46 053510 · Zbl 1110.37055
[5] Kalnins E G, Kress J M and Miller W Jr 2005 III. 3D classical structure theory J. Math. Phys.46 103507 · Zbl 1111.37055
[6] Kalnins E G, Kress J M and Miller W Jr 2006 IV. The classical 3D Stäckel transform and 3D classification theory J. Math. Phys.47 043514 · Zbl 1112.37058
[7] Kalnins E G, Kress J M and Miller W Jr 2006 V: 2D and 3D quantum systems J. Math. Phys.47 09350
[8] Kalnins E G, Kress J M and Miller W Jr 2007 Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties J. Phys. A: Math. Theor.40 3399-411 · Zbl 1128.37037
[9] Kalnins E G, Kress J M, Miller W Jr and Post S 2011 Laplace-type equations as conformal superintegrable systems Adv. Appl. Math.46 396416 · Zbl 1268.37078
[10] Capel J J, Kress J M and Post S 2015 Invariant classification and limits of maximally superintegrable systems in 3D SIGMA11 038 · Zbl 1318.33014
[11] Kalnins E G, Miller W Jr and Subag E 2016 Laplace equations, conformal superintegrability and Bôcher contractions Acta Polytech.56 214-23
[12] Kalnins E G, Miller W Jr and Subag E 2016 Bôcher contractions of conformally superintegrable Laplace equations SIGMA12 038 · Zbl 1338.81228
[13] Capel J J and Kress J M 2014 Invariant classification of second-order conformally flat superintegrable systems J. Phys. A: Math. Theor.47 495202 · Zbl 1314.14064
[14] Evans N W 1990 Superintegrability in classical mechanics Phys. Rev. A 41 5668-70
[15] Kalnins E G, Kress J M and Miller W Jr 2007 Fine structure for 3D second order superintegrable systems: 3-parameter potentials J. Phys. A: Math. Theor.40 5875-92 · Zbl 1114.37032
[16] Verrier P E and Evans N W 2008 A new superintegrable Hamiltonian J. Math. Phys.49 022902 · Zbl 1153.81446
[17] Tanoudis Y and Daskaloyannis C 2011 Algebraic calculation of the energy eigenvalues for the nondegenerate three-dimensional Kepler-Coulomb potential SIGMA7 054 · Zbl 1217.81108
[18] Kalnins E G, Kress J M and Miller W Jr 2013 Extended Kepler-Coulomb quantum superintegrable systems in 3 dimensions J. Phys. A: Math. Theor.46 085206 · Zbl 1264.81183
[19] Rodriguez M A, Tempesta P and Winternitz P 2008 Reduction of superintegrable systems: the anisotropic harmonic oscillator Phys. Rev. E 78 046608
[20] Evans N W and Verrier P E 2008 Superintegrability of the caged anisotropic oscillator J. Math. Phys.49 092902 · Zbl 1152.81424
[21] Escobar-Ruiz M A and Miller W Jr 2016 Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces J. Phys. A: Math. Theor.49 305202 · Zbl 1346.81050
[22] Inönü E and Wigner E P 1953 On the contraction of groups and their representations Proc. Natl Acad. Sci.39 510-24 · Zbl 0050.02601
[23] Kalnins E G and Miller W Jr 2014 Quadratic algebra contractions and 2nd order superintegrable systems Anal. Appl.12 583-612 · Zbl 1300.22013
[24] Bôcher M 1894 Ueber die Reihenentwickelungen der Potentialtheorie (Leipzig: B G Teubner)
[25] Escobar-Ruiz M A, Kalnins E G, Miller W Jr and Subag E 2016 Bôcher and abstract contractions of 2nd order quadratic algebras (arXiv:1611.02560 [math-ph]) (submitted)
[26] Gantmacher F R 1960 Theory of Matrices vol 2 (New York: Chelsea) (translation of the Russian original)
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