PoS - Proceedings of Science
Volume 292 - Corfu Summer Institute 2016 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2016) - Workshop on Geometry and Physics, Ringberg Castle, 20-25 November 2016, invited contributions
Calogero, spherically reduced and PT-deformed
F. Correa and O. Lechtenfeld*
Full text: pdf
Pre-published on: October 04, 2017
Published on: October 05, 2017
The rational Calogero model based on the $A_{n-1}$ root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n-2}$ subject to a very particular potential singular at the Weyl chamber walls. We review the computation of its energy spectrum (including the eigenstates),
conserved charges and intertwining operators shifting the coupling constant by one. These models are deformed 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 levels but in some cases adds a previously unphysical tower of states. For integral couplings this roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry-type charge.
We illustrate the general constructions by presenting the details for the cases of $A_2$ and $A_3$ and point out open questions.
DOI: https://doi.org/10.22323/1.292.0095
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