Abstract
In this short review we describe the integrability properties of the Calogero-type perturbations of one- and two-center Coulomb problems and of the Stark–Coulomb problem. We present the explicit expressions of their constants of motion and show that these systems admit partial separation of variables.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Komarov and S. Slavyanov, “The two Coulomb centres problem at large centre separation,” J. Phys. B 1, 1066 (1968)
Yu. Demkov and I. Komarov, “Hypergeometric partial solutions in the problem of two Coulomb centers,” Theor. Math. Phys. 38, 174 (1979).
F. Calogero, “Solution of a three-body problem in one dimension,” J. Math. Phys. 10 (1969); “Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials,” J. Math. Phys. 12, 419 (1971).
S. Wojciechowski, “Superintegrability of the Calogero-Moser system,” Phys. Lett. A 95, 279 (1983).
J. Moser, “Three integrable hamiltonian systems connected with isospectral deformations,” Adv. Math. 16, 197 (1975).
M. Olshanetsky and A. Perelomov, “Classical integrable finite dimensional systems related to lie algebras,” Phys. Rep. 71, 313 (1981).
A. P. Polychronakos, “Physics and mathematics of Calogero particles,” J. Phys. A 39, 12793 (2006).
A. Khare, “Exact solution of an N-body problem in one dimension,” J. Phys. A 29, L45 (1996).
T. Hakobyan, O. Lechtenfeld, and A. Nersessian, “Superintegrability of generalized Calogero models with oscillator or Coulomb potential,” Phys. Rev. D: Part. Fields 90, 101701(R) (2014).
T. Hakobyan and A. Nersessian, “Runge-Lenz vector in Calogero-Coulomb problem,” Phys. Rev. A 92, 022111 (2015).
T. Hakobyan and A. Nersessian, “Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems,” Phys. Rev. D: Part. Fields 93, 045025 (2016).
M. Feigin and T. Hakobyan, “On dunkl angular momenta algebra,” J. High Energy Phys. 11, 107 (2015).
M. Feigin, O. Lechtenfeld, and A. Polychronakos, “The quantum angular Calogero-Moser model,” J. High Energy Phys. 1307, 162 (2013).
T. Hakobyan, A. Nersessian, and V. Yeghikyan, “Cuboctahedric higgs oscillator from the Calogero model,” J. Phys. A 42, 205206 (2009).
F. Correa and O. Lechtenfeld, “The tetrahexahedric angular Calogero model,” J. High Energy Phys. 10, 191 (2015).
T. Hakobyan, D. Karakhanyan, and O. Lechtenfeld, “The structure of invariants in conformal mechanics,” Nucl. Phys. B 886, 399 (2014)
T. Hakobyan, S. Krivonos, O. Lechtenfeld, and A. Nersessian, “Hidden symmetries of integrable conformal mechanical systems,” Phys. Lett. A 374, 801 (2010)
T. Hakobyan, O. Lechtenfeld, A. Nersessian, and A. Saghatelian, “Invariants of the spherical sector in conformal mechanics,” J. Phys. A 44, 055205 (2011)
O. Lechtenfeld, A. Nersessian, and V. Yeghikyan, “Action-angle variables for dihedral systems on the circle,” Phys. Lett. A 374, 4647 (2010).
C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Trans. Am. Math. Soc. 311, 167 (1989).
M. Feigin, “Intertwining relations for the spherical parts of generalized Calogero operators,” Theor. Math. Phys. 135, 497 (2003).
V. X. Genest, A. Lapointe, and L. Vinet, “The Dunkl-Coulomb problem in the plane,” Phys. Lett. A 379, 923 (2015).
T. Hakobyan, O. Lechtenfeld, and A. Nersessian, “The spherical sector of the Calogero model as a reduced matrix model,” Nucl. Phys. B 858, 250 (2012).
P. J. Redmond, “Generalization of the Runge-Lenz vector in the presence of an electric field,” Phys. Rev. B 133, 1352 (1964).
K. Helfrich, “Constants of motion for separable oneparticle problems with cylinder symmetry,” Theor. Chim. Acta (Berl.) 24, 271 (1972).
L. Landau and E. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1989; Pergamon, Oxford, 1977).
C. Coulson and A. Joseph, “A constant of the motion for the two-centre Kepler problem,” Int. J. Quantum Chem. 1, 337 (1967).
H. A. Erikson and E. L. Hill, “A note about one-electron states of diatomic molecules,” Phys. Rev. 76, 29 (1949).
S. P. Alliluev and A. V. Matveenko, “Symmetry group of the hydrogen molecular ion (A system with separable variables),” Sov. Phys. JETP 24, 1260 (1967).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article is published in the original.
Rights and permissions
About this article
Cite this article
Hakobyan, T., Nersessian, A. Integrability of Calogero–Coulomb problems. Phys. Part. Nuclei Lett. 14, 331–335 (2017). https://doi.org/10.1134/S1547477117020133
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1547477117020133