
SIGMA 1 (2005), 015, 17 pages nlin.SI/0511035
https://doi.org/10.3842/SIGMA.2005.015
Second Order Superintegrable Systems in Three Dimensions
Willard Miller
School of Mathematics, University of Minnesota,
Minneapolis, Minnesota, 55455, USA
Received October 28, 2005; Published online November 13, 2005
Abstract
A classical (or quantum) superintegrable system on an
ndimensional Riemannian manifold is an integrable
Hamiltonian system with potential that admits 2n1
functionally independent constants of the motion that are
polynomial in the momenta, the maximum number possible.
If these constants of the motion are all quadratic, the system is second order superintegrable.
Such systems have remarkable properties. Typical properties are that
1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems,
2) they are multiseparable, 3) the second order symmetries generate a
closed quadratic algebra and in the quantum case the representation
theory of the quadratic algebra yields important facts about the
spectral resolution of the Schrödinger operator and the other
symmetry operators, and 4) there are deep
connections with expansion formulas relating classes of special
functions and with the theory of Exact and Quasiexactly Solvable systems.
For n = 2 the author, E.G. Kalnins and J. Kress,
have worked out the structure of these systems and classified all of the possible spaces and potentials.
Here I discuss our recent work and announce new results for the much more difficult case
n = 3. We
consider classical superintegrable systems with nondegenerate
potentials in three dimensions and on a conformally flat real or complex space.
We show that there exists a standard structure for such systems, based on the algebra
of 3×3 symmetric matrices,
and that the quadratic algebra always closes at order 6. We describe the
Stäckel transformation, an invertible conformal mapping between
superintegrable structures on distinct spaces, and give evidence indicating that all
our superintegrable systems are Stäckel transforms
of systems on complex Euclidean space or the complex 3sphere.
We also indicate how to
extend the classical 2D and 3D superintegrability
theory to include the operator (quantum) case.
Key words:
superintegrability; quadratic algebra; conformally flat spaces.
pdf (246 kb)
ps (190 kb)
tex (21 kb)
References

Wojciechowski S.,
Superintegrability of the CalogeroMoser system,
Phys. Lett. A, 1983, V.95, 279281.
 Evans N.W.,
Superintegrability in classical mechanics,
Phys. Rev. A, 1990, V.41, 56665676; Group theory of the
SmorodinskyWinternitz system, J. Math. Phys., 1991, V.32, 33693375.
 Evans N.W.,
Superintegrability of the Winternitz system,
Phys. Lett. A, 1990, V.147, 483486.
 Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P.,
On higher symmetries in quantum mechanics,
Phys. Lett., 1965, V.16, 354356.
 Fris J., Smorodinskii Ya.A., Uhlír M., Winternitz P.,
Symmetry groups in classical and quantum mechanics,
Sov. J. Nucl. Phys., 1967, V.4, 444450.
 Makarov A.A., Smorodinsky Ya.A., Valiev Kh.,
Winternitz P.,
A systematic search for nonrelativistic systems with
dynamical symmetries,
Nuovo Cimento, 1967, V.52, 10611084.
 Calogero F.,
Solution of a threebody problem in one
dimension, J. Math. Phys., 1969, V.10, 21912196.
 Cisneros A., McIntosh H.V.,
Symmetry of the
twodimensional hydrogen atom,
J. Math. Phys., 1969, V.10, 277286.
 Sklyanin E.K.,
Separation of variables in the Gaudin model,
J. Sov. Math., 1989, V.47, 24732488.
 Faddeev L.D., Takhtajan L.A.,
Hamiltonian methods in the theory of solitons, Berlin,
Springer, 1987.
 Harnad J.,
Loop groups, Rmatrices and separation of variables,
in "Integrable Systems: From Classical to Quantum", Editors J. Harnad, G. Sabidussi and P. Winternitz,
CRM Proceedings and Lecture Notes, 2000, V.26, 2154.
 Eisenhart L.P.,
Riemannian geometry, Princeton University Press, 2^{nd} printing,
1949.
 Miller W.Jr.,
Symmetry and separation of variables, Providence, Rhode Island,
AddisonWesley Publishing Company, 1977.
 Kalnins E.G., Miller W.Jr.,
Killing tensors and variable separation for HamiltonJacobi and Helmholtz equations,
SIAM J. Math. Anal., 1980, V.11, 10111026.
 Miller W.,
The technique of variable separation for partial differential equations,
in Proceedings of School and Workshop on Nonlinear Phenomena (November 29  December 17, 1982,
Oaxtepec, Mexico), Lecture Notes in Physics, Vol. 189,
New York, SpringerVerlag, 1983, 184208.
 Kalnins E.G.,
Separation of variables for Riemannian spaces of constant
curvature, Pitman, Monographs and Surveys in Pure and Applied Mathematics,
Vol. 28, Essex, England,
Longman, 1986, 184208,
 Miller W.Jr.,
Mechanisms for variable separation in partial
differential equations and their relationship to group theory, in
"Symmetries and Nonlinear Phenomena", World Scientific,
1988, 188221
 Kalnins E.G., Kress J.M., Miller W.Jr.,
Secondorder superintegrable systems in conformally flat spaces.
I. Twodimensional classical structure theory, J. Math. Phys., 2005,
V.46, 053509, 28 pages.
 Kalnins E.G., Kress J.M., Miller W.Jr.,
Second order superintegrable systems in conformally flat spaces. II. The classical
twodimensional Stäckel transform, J. Math. Phys.,
2005, V.46, 053510, 15 pages.
 Kalnins E.G., Kress J.M., Miller W.Jr.,
Second order superintegrable systems in conformally flat spaces. III.
Threedimensional classical structure theory,
J. Math. Phys., 2005, V.46, 103507, 28 pages.
 Kalnins E.G., Kress J.M., Miller W.Jr.,
Second order superintegrable systems in conformally
flat spaces. IV. The classical
threedimensional Stäckel transform, submitted.
 Kalnins E.G., Miller W.Jr., Pogosyan G.S.,
Superintegrability in three dimensional Euclidean space,
J. Math. Phys., 1999, V.40, 708725.
 Kalnins E.G., Miller W.Jr., Pogosyan G.S.,
Superintegrability and associated polynomial solutions. Euclidean space and
the sphere in two dimensions,
J. Math. Phys., 1996, V.37, 64396467.
 Bonatos D., Daskaloyannis C., Kokkotas K.,
Deformed
oscillator algebras for twodimensional quantum superintegrable
systems,
Phys. Rev. A, 1994, V.50, 37003709, hepth/9309088.

Daskaloyannis C.,
Quadratic Poisson algebras of twodimensional classical
superintegrable systems and quadratic associate algebras of quantum
superintegrable systems,
J. Math. Phys., 2001, V.42, 11001119, mathph/0003017.

Smith S.P.,
A class of algebras similar to the enveloping algebra of sl(2),
Trans. Amer. Math. Soc., 1990, V.322, 285314.

Kalnins E.G., Miller W., Tratnik M.V.,
Families of orthogonal and
biorthogonal polynomials on the nsphere,
SIAM J. Math. Anal., 1991, V.22, 272294.
 Ushveridze A.G.,
Quasiexactly solvable models in quantum mechanics,
Bristol, Institute of Physics, 1993.
 Letourneau P., Vinet L.,
Superintegrable systems:
polynomial algebras and quasiexactly solvable
Hamiltonians, Ann. Phys., 1995, V.243, 144168.
 Kalnins E.G., Miller W.Jr., Pogosyan G.S.,
Exact and quasiexact solvability of second order superintegrable
systems. I. Euclidean space preliminaries, submitted.
 Grosche C., Pogosyan G.S., Sissakian A.N.,
Path integral
discussion for SmorodinskyWinternitz potentials: I. Two and threedimensional Euclidean space,
Fortschritte der Physik, 1995, V.43,
453521, hepth/9402121.
 Kalnins E.G., Kress J.M., Miller W.Jr., Pogosyan G.S.,
Completeness of superintegrability in twodimensional constant
curvature spaces,
J. Phys. A: Math. Gen., 2001, V.34, 47054720, mathph/0102006.
 Kalnins E.G., Kress J.M., Winternitz P.,
Superintegrability in a twodimensional space of nonconstant curvature,
J. Math. Phys., 2002, V.43, 970983, mathph/0108015.
 Kalnins E.G., Kress J.M., Miller W.Jr., Winternitz P.,
Superintegrable systems in Darboux spaces,
J. Math. Phys., 2003, V.44, 58115848, mathph/0307039.

Rañada M.F.,
Superintegrable n=2 systems, quadratic constants of motion, and
potentials of Drach,
J. Math. Phys., 1997, V.38, 41654178.

Kalnins E.G., Miller W.Jr., Williams G.C., Pogosyan G.S.,
On superintegrable symmetrybreaking potentials in
ndimensional Euclidean space,
J. Phys. A: Math. Gen., 2002, V.35, 46554720.
 Boyer C.P., Kalnins E.G., Miller W.,
Stäckelequivalent integrable Hamiltonian systems,
SIAM J. Math. Anal., 1986, V.17, 778797.
 Hietarinta J., Grammaticos B., Dorizzi B., Ramani A.,
Couplingconstant metamorphosis and duality between
integrable Hamiltonian systems,
Phys. Rev. Lett., 1984, V.53, 17071710.
 Kalnins E.G., Miller W., Reid G.K.,
Separation of variables for Riemannian spaces of constant
curvature. I. Orthogonal separable coordinates for S_{c} and E_{nC},
Proc. R. Soc. Lond. A, 1984, V.39, 183206.

