scholarly journals Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

2006 ◽  
Vol 47 (4) ◽  
pp. 043514 ◽  
Author(s):  
E. G. Kalnins ◽  
J. M. Kress ◽  
W. Miller
Keyword(s):  
Second Order ◽  
Classification Theory ◽  
Conformally Flat ◽  
Superintegrable Systems

scholarly journals Quadratic algebra contractions and second-order superintegrable systems

2014 ◽  
Vol 12 (05) ◽  
pp. 583-612 ◽  
Author(s):  
Ernest G. Kalnins ◽  
W. Miller
Keyword(s):  
Constant Curvature ◽  
Two Dimensions ◽  
Second Order ◽  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Physical Systems ◽  
Special Cases ◽  
Superintegrable Systems ◽  
Wilson Polynomials ◽  
Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

scholarly journals Superintegrable systems with spin and second-order integrals of motion

2012 ◽  
Vol 45 (47) ◽  
pp. 475201 ◽  
Author(s):  
Jean-Francois Désilets ◽  
Pavel Winternitz ◽  
İsmet Yurduşen
Keyword(s):  
Second Order ◽  
Integrals Of Motion ◽  
Superintegrable Systems
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