English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We show explicitly that all 2nd order superintegrable systems in 2 dimensionsare limiting cases of a single system: the generic 3-parameter potential on the2-sphere, S9 in our listing. We extend the Wigner-In\"on\"u method of Liealgebra contractions to contractions of quadratic algebras and show that all ofthe quadratic symmetry algebras of these systems are contractions of that ofS9. Amazingly, all of the relevant contractions of these superintegrablesystems on flat space and the sphere are uniquely induced by the well known Liealgebra contractions of e(2) and so(3). By contracting function spacerealizations of irreducible representations of the S9 algebra (which give thestructure equations for Racah/Wilson polynomials) to the other superintegrablesystems, and using Wigner's idea of "saving" a representation, we obtain thefull Askey scheme of hypergeometric orthogonal polynomials. This relationshipdirectly ties the polynomials and their structure equations to physicalphenomena. It is more general because it applies to all special functions thatarise from these systems via separation of variables, not just those ofhypergeometric type, and it extends to higher dimensions.

Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials