The tetrahexahedric angular Calogero model

Francisco Correa,Olaf Lechtenfeld

doi:10.1007/jhep10(2015)191

Francisco Correa, Olaf Lechtenfeld

Open Access

https://doi.org/10.1007/jhep10(2015)191

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Abstract

The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

Highlights

JHEP10(2015)191 where A and B are polynomial in C′ and linear in C′′, and we have suppressed the counting labels
The spherical reduction of the rational Calogero model is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential
Since the rational Calogero Hamiltonian is part of an SL(2,R) conformal algebra, it is natural to reduce the rational model to an integrable system on a sphere

Summary

Warmup: the hexagonal or Poschl-Teller model

Its angular cousin take the following form as differential operators on R2 and S1, respectively,. We display the angular Hamiltonian in complex coordinates as well, HΩ =. The angular wave function reads vq(φ) ≡ vl(g)(φ) ∼ rq 3 Dν3 l3 ∆g r−6g ∼ rq Dw3 − Dw3 ̄ l3 ∆g r−6g ν=1. Owing to the S3 Weyl group of A2, we take profit from the complex cubic roots of unity to cast the holomorphic Dunkl operator. Where the three basic Weyl reflections act as follows in the complex plane, s0 : w → −w , s+ : w → −ρw , s− : w → −ρw ,. With the abbreviated notation xμν ≡ xμ−xν and ∂μν ≡ ∂xμ−∂xν For generic values of g, there can be no further independent conserved quantity.

Tetrahexahedric model: the spectrum

Dμ3 l3

Outlook

A Complementary formulae