Abstract
The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of \( {{\mathbb{R}}^n} \) to the ‘angular Calogero-Moser model’ on the sphere S n−1.We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a ‘relative angular Calogero-Moser model’, which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.
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References
F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].
M. Olshanetsky and A. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].
M. Olshanetsky and A. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].
A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].
Calogero-Moser system Scholarpedia webpage, http://www.scholarpedia.org/article/Calogero-Moser_system.
T. Hakobyan, A. Nersessian and V. Yeghikyan, Cuboctahedric Higgs oscillator from the Calogero model, J. Phys. A 42 (2009) 205206 [arXiv:0808.0430] [INSPIRE].
T. Hakobyan, S. Krivonos, O. Lechtenfeld and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374 (2010) 801 [arXiv:0908.3290] [INSPIRE].
T. Hakobyan, O. Lechtenfeld, A. Nersessian and A. Saghatelian, Invariants of the spherical sector in conformal mechanics, J. Phys. A 44 (2011) 055205 [arXiv:1008.2912] [INSPIRE].
T. Hakobyan, O. Lechtenfeld and A. Nersessian, The spherical sector of the Calogero model as a reduced matrix model, Nucl. Phys. B 858 (2012) 250 [arXiv:1110.5352] [INSPIRE].
O. Lechtenfeld, A. Nersessian and V. Yeghikyan, Action-angle variables for dihedral systems on the circle, Phys. Lett. A 374 (2010) 4647 [arXiv:1005.0464] [INSPIRE].
T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian and V. Yeghikyan, Action-angle variables and novel superintegrable systems, Phys. Part. Nuclei 43 (2012) 577 [INSPIRE].
M.V. Feigin, Intertwining relations for spherical parts of generalized Calogero operators, Theor. Math. Phys. 135 (2003) 497.
J.A. Minahan and A.P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993) 265 [hep-th/9206046] [INSPIRE].
F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].
F. Calogero and C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials, J. Math. Phys. 15 (1974) 1425 [INSPIRE].
P.W. Higgs, Dynamical symmetries in a spherical geometry. 1, J. Phys. A 12 (1979) 309 [INSPIRE].
H.I. Leemon, Dynamical symmetries in a spherical geometry. 2, J. Phys. A 12 (1979) 489 [INSPIRE].
A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].
L. Brink, T. Hansson and M.A. Vasiliev, Explicit solution to the N body Calogero problem, Phys. Lett. B 286 (1992) 109 [hep-th/9206049] [INSPIRE].
C.F. Dunkl and Y. Hu, Orthogonal polynomials of several variables, Cambridge University Press, Cambridge U.K. (2001).
J.F. Van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Commun. Math. Phys. 188 (1997) 467 [q-alg/9609032].
G.J. Heckman, A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, W. Barker and P. Sally eds., Progr. Math. 101 (1991) 181, Birkhäuser, (1991).
C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994) 237.
J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge U.K. (1990).
J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974) 1420 [INSPIRE].
M. Olshanetsky, Wave functions of quantum integrable systems, Theor. Math. Phys. 57 (1983) 1048 [Teor. Mat. Fiz. 57 (1983) 148] [INSPIRE].
A. Golynski, diploma paper, Moscow State University, Moscow Russia (1999).
A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 [hep-th/9210109] [INSPIRE].
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Feigin, M., Lechtenfeld, O. & Polychronakos, A.P. The quantum angular Calogero-Moser model. J. High Energ. Phys. 2013, 162 (2013). https://doi.org/10.1007/JHEP07(2013)162
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DOI: https://doi.org/10.1007/JHEP07(2013)162