$\mathcal{P}\mathcal{T}$ deformation of angular Calogero models

Abstract

The rational Calogero model based on an arbitrary rank-$n$ Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n-1}$ subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a $\mathcal{P}\mathcal{T}$-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The $\mathcal{P}\mathcal{T}$ deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two ($A_2$, $G_2$) and all rank-three Coxeter systems ($A{D}_3$, $B{C}_3$ and $H_3$), including a reducible case ($A_1^{\otimes 3}$).

arXiv comment1+41 pages, 12 figures

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• $\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models
• Integrability, intertwiners and non-linear algebras in Calogero models
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Coupling and particle number intertwiners in the Calogero model