(Quasi)-exact-solvability on the sphere Sn

Willard Miller, Alexander V. Turbiner

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

An Exactly Solvable (ES) potential on the sphere Sn is reviewed and a related Quasi-Exactly Solvable (QES) potential is found and studied. After mapping the sphere to a simplex, it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is gln, realized in a finite-dimensional representation by first order differential operators acting on RPn. It is shown that variables in the Schrödinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability. The QES system is completely integrable for n = 2 and non-maximally superintegrable for n ≥ 3. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Inönü-Wigner Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our Sn system to a QES non-maximal superintegrable system on Euclidean space En, an extension of the Smorodinsky-Winternitz potential.

Original languageEnglish (US)
Article number023501
JournalJournal of Mathematical Physics
Volume56
Issue number2
DOIs
StatePublished - Feb 3 2015

Bibliographical note

Publisher Copyright:
© 2015 AIP Publishing LLC.

Fingerprint

Dive into the research topics of '(Quasi)-exact-solvability on the sphere Sn'. Together they form a unique fingerprint.

Cite this