Quantum integrals from coalgebra structure

Quantum versions of the hydrogen atom and the harmonic oscillator are studied on non Euclidean spaces of dimension N. 2 N − 1 ?> integrals, of arbitrary order, are constructed via a multi-dimensional version of the factorization method, thus confirming the conjecture of Riglioni (2013 J. Phys. A: Math. Theor. 46 265207). The systems are extended via coalgebra extension of sl ( 2 ) ?> representations, although not all integrals are expressible in these generators. As an example, dimensional reduction is applied to four-dimensional systems to obtain extension and new proofs of the superintegrability of known families of Hamiltonians.

[1]  R. Minlos,et al.  A system of three quantum particles with point-like interactions , 2014 .

[2]  P. Winternitz,et al.  Superintegrable systems with spin induced by co-algebra symmetry , 2013, 1312.3273.

[3]  W. Miller,et al.  Classical and quantum superintegrability with applications , 2013, 1309.2694.

[4]  D. Riglioni.  Classical and quantum higher order superintegrable systems from coalgebra symmetry , 2013, 1304.4918.

[5]  A. Enciso,et al.  Superintegrable quantum oscillator and Kepler-Coulomb systems on curved spaces , 2013, 1304.4544.

[6]  J. Negro,et al.  A unified approach to quantum and classical TTW systems based on factorizations , 2012 .

[7]  O. Lechtenfeld,et al.  Integrable generalizations of oscillator and Coulomb systems via action–angle variables , 2011, 1108.5189.

[8]  C. Gonera.  On the superintegrability of TTW model , 2010, 1010.2915.

[9]  I. Marquette.  Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems , 2011, 1109.4471.

[10]  W. Miller,et al.  Superintegrability and higher-order constants for classical and quantum systems , 2010, 1002.2665.

[11]  S. Post.  Coupling constant metamorphosis, the Stäckel transform and superintegrability , 2010 .

[12]  O. Lechtenfeld,et al.  Action-angle variables for dihedral systems on the circle , 2010, 1005.0464.

[13]  P. Winternitz,et al.  An infinite family of superintegrable deformations of the Coulomb potential , 2010, 1003.5230.

[14]  W. Miller,et al.  Superintegrability and higher order integrals for quantum systems , 2009, 0912.2278.

[15]  C. Quesne.  Superintegrability of the Tremblay–Turbiner–Winternitz quantum Hamiltonians on a plane for odd k , 2009, 0911.4404.

[16]  W. Miller,et al.  Coupling constant metamorphosis and Nth-order symmetries in classical and quantum mechanics , 2009, 0908.4393.

[17]  I. Marquette.  Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion , 2009, 0908.1246.

[18]  P. Tempesta,et al.  Symmetry reduction and superintegrable Hamiltonian systems , 2009, 0906.3396.

[19]  P. Winternitz,et al.  An infinite family of solvable and integrable quantum systems on a plane , 2009, 0904.0738.

[20]  A. Hinze,et al.  Second order superintegrable systems in conformally flat spaces . 2 : The classical 2 D Stäckel transform , 2009 .

[21]  A. Enciso,et al.  Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds , 2008, 0810.0999.

[22]  P. Tempesta,et al.  Reduction of superintegrable systems: the anisotropic harmonic oscillator. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  A. Enciso,et al.  N-dimensional sl(2)-coalgebra spaces with non-constant curvature , 2007, 0704.1470.

[24]  F. J. Herranz,et al.  Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature , 2006, math-ph/0610040.

[25]  W. Miller,et al.  Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform , 2005 .

[26]  C. Quesne,et al.  Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem , 2004, math-ph/0403047.

[27]  Y. Kurochkin,et al.  Model of excitations in quantum dots based on quantum mechanics in spaces of constant curvature , 2001 .

[28]  W. Miller,et al.  Completeness of superintegrability in two-dimensional constant-curvature spaces , 2001, math-ph/0102006.

[29]  P. Horvathy,et al.  Dynamical symmetry of monopole scattering , 1987 .

[30]  B. Dorizzi,et al.  Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems , 1984 .