Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

E. G. Kalnins, J. M. Kress, W. Miller

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

A classical (or quantum) second order superintegrable system is an integrable n -dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials.

Original languageEnglish (US)
Article number113518
JournalJournal of Mathematical Physics
Volume48
Issue number11
DOIs
StatePublished - 2007

Fingerprint Dive into the research topics of 'Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties'. Together they form a unique fingerprint.

Cite this