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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

2nd-order conformal superintegrable systems in $n$ dimensions are Laplaceequations on a manifold with an added scalar potential and $2n - 1$ independent2nd order conformal symmetry operators. They encode all the information aboutHelmholtz (eigenvalue) superintegrable systems in an efficient manner: there isa 1-1 correspondence between Laplace superintegrable systems and Stackelequivalence classes of Helmholtz superintegrable systems. In this paper wefocus on superintegrable systems in two dimensions, $n = 2$, where there are 44Helmholtz systems, corresponding to 12 Laplace systems. For each Laplaceequation we determine the possible 2-variate polynomial subspaces that areinvariant under the action of the Laplace operator, thus leading to families ofpolynomial eigenfunctions. We also study the behavior of the polynomialinvariant subspaces under a Stackel transform. The principal new results arethe details of the polynomial variables and the conditions on parameters of thepotential corresponding to polynomial solutions. The hidden gl_3-algebraicstructure is exhibited for the exact and quasi-exact systems. For physicallymeaningful solutions, the orthogonality properties and normalizability of thepolynomials are presented as well. Finally, for all Helmholtz superintegrablesolvable systems we give a unified construction of 1D and 2D quasi-exactlysolvable potentials possessing polynomial solutions, and a construction of new2D PT-symmetric potentials is established.

Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces