Non-Hermitian superintegrable systems

Abstract

A non-Hermitian generalisation of the Marsden--Weinstein reduction method is introduced to construct families of quantum $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-symmetric breaking. In certain limits, the models reduce to known non-Hermitian systems and complex extensions of previously studied real superintegrable systems.

18 pages, no figures

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