Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

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dc.contributor.authorKalnins, Ernie G.
dc.contributor.authorKress, Jonathan M.
dc.contributor.authorMiller, W., Jr.
dc.date.accessioned2008-10-29T01:23:50Z
dc.date.available2008-10-29T01:23:50Z
dc.date.issued2005-04
dc.description.abstractThis paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2×2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.en_US
dc.format.mimetypeapplication/pdf
dc.identifier.citationKalnins, E.G., Kress, J.M. & Miller, W., Jr. (2005). Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory. Journal of Mathematical Physics, 46, 053509.en_US
dc.identifier.doi10.1063/1.1897183en_US
dc.identifier.issn0022-2488
dc.identifier.urihttps://hdl.handle.net/10289/1179
dc.language.isoen
dc.relation.isPartOfJournal of Mathematical Physicsen_NZ
dc.relation.urihttp://link.aip.org/link/?JMAPAQ/46/053509/1en_US
dc.rightsCopyright 2005 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in the Journal of Mathematical Physics and may be found at http://jmp.aip.org/jmp/top.jspen_US
dc.subjectMathematicsen_US
dc.subjectSchrodinger equationen_US
dc.subjectquantum theoryen_US
dc.subjectmatrix algebraen_US
dc.subjectmathematical operatorsen_US
dc.titleSecond-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theoryen_US
dc.typeJournal Articleen_US
pubs.begin-page053509en_NZ
pubs.elements-id84065
pubs.end-page053509en_NZ
pubs.issue5en_NZ
pubs.volume46en_NZ
uow.identifier.article-no053509en_NZ
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