Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries

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dc.contributor.authorKalnins, Ernie G.
dc.contributor.authorMiller, W., Jr.
dc.contributor.authorPogosyan, G.S.
dc.date.accessioned2008-10-29T00:37:27Z
dc.date.available2008-10-29T00:37:27Z
dc.date.issued2006-03
dc.description.abstractWe show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set.en_US
dc.format.mimetypeapplication/pdf
dc.identifier.citationKalnins, E.G., Miller, W., Jr. & Pogosyan, G.S. (2006). Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries. Journal of Mathematical Physics, 47, 033502 .en_US
dc.identifier.doi10.1063/1.2174237en_US
dc.identifier.issn0022-2488
dc.identifier.urihttps://hdl.handle.net/10289/1175
dc.language.isoen
dc.publisherAmerican Institute of Physicsen_NZ
dc.relation.isPartOfJournal of Mathematical Physicsen_NZ
dc.relation.urihttp://link.aip.org/link/?JMAPAQ/48/023503/1en_US
dc.rightsCopyright 2006 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.subjectquantum theoryen_US
dc.subjectpolynomialsen_US
dc.titleExact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminariesen_US
dc.typeJournal Articleen_US
pubs.begin-page033502en_NZ
pubs.elements-id32715
pubs.end-page033502en_NZ
pubs.issue3en_NZ
pubs.volume47en_NZ
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