Non-Hermitian superintegrable systems

Abstract

A non-Hermitian generalisation of the Marsden--Weinstein reduction method is introduced to construct families of quantum \(\mathcal{PT}\)-symmetric superintegrable models over an \(n\)-dimensional sphere \(S^n\). The mechanism is illustrated with one- and two-dimensional examples, related to \(u(2)\) and \(u(3)\) Lie algebras respectively, providing new quantum models with real spectra and spontaneous \(\mathcal{PT}\)-symmetric breaking. In certain limits, the models reduce to known non-Hermitian systems and complex extensions of previously studied real superintegrable systems.