# ${\cal PT}$ deformation of angular Calogero models

## TL;DR

This paper explores ${
m PT}$-symmetric deformations of angular Calogero models, revealing how such transformations affect the potential, energy spectrum, and symmetries, especially for specific Coxeter systems.

## Contribution

It introduces a ${
m PT}$-symmetric deformation method for angular Calogero models, showing how to modify potentials and analyze spectral and symmetry properties for various Coxeter systems.

## Key findings

- ${
m PT}$ deformation preserves energy eigenvalues.
- Deformation can add unphysical states and double degeneracy.
- For integral couplings, energy levels coincide, enabling nonlinear supersymmetry.

## 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 ${\cal PT}$-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The ${\cal PT}$ 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 ($AD_3$, $BC_3$ and $H_3$), including a reducible case ($A_1^{\otimes 3}$).