Abstract
We point out some directions for potential generalizations of Calogero-type systems. In particular, we demonstrate that a many-matrix model gives rise, upon Hamiltonian reduction, to a multidimensional version of the Calogero—Sutherland—Moser model and its spin generalizations. Some simple solutions of these models are demonstrated by solving the corresponding matrix equations. We also show that a supersymmetric system of spinless particles in which supersymmetry is realized through exchange operators exhibits reflectionless two-body scattering for arbitrary prepotential. The exchange-Calogero system is the simplest example, but it is conjectured that appropriate three-body forces would make all such systems integrable.
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References
L. Brink, T. H. Hansson, and M. A. Vassiliev, Explicit solution to the N-body Calogero problem, Phys. Lett. B 286 (1992), No. 1-2, 109–11.
F. Calogero, Ground state of a one-dimensional N-body system, J. Math. Phys. 10 (1969), 2197–2200.
F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191–2196.
F. Calogero, Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419–436.
F. Calogero, Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cimento 13 (1975), No. 11, 411–416.
C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), No. 1, 167–183.
J. Gibbons and T. Hermsen, A generalisation of the Calogero-Moser system, Physica 11D (1984), No. 3, 337–348.
A. Gorsky and N. Nekrasov, Quantum integrable systems of particles as gauge theories, Theoret, and Math. Phys. 100 (1994), No. 1, 874–878.
Z. N. C. Ha and F. D. M. Haldane, On models with inverse-square exchange, Phys. Rev. B 46 (1992), 9359–9368.
D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Commun. Pure Appi. Math. 31 (1978), No. 4, 481–507.
L. Lapointe and L. Vinet, Exact operator solution of the Calo-gero-Sutherland model, Commun. Math. Phys. 178 (1996), No. 2, 425–452, hep-th/9507073.
C. Marchioro, F. Calogero, and O. Ragnisco, Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential V a(x) = g 2 a 2/sinh2(ax), Lett. Nuovo Cimento 13 (1975), No. 10, 383–387.
J. A. Minahan and A. P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993), No. 2-3, 265–270.
J. A. Minahan and A. P. Polychronakos, Interacting fermion systems from two-dimensional QCD, Phys. Lett. B 336 (1994), 288–294.
J. Moser, Three integrable Hamiltonian systems connected to isospec-tral deformations, Adv. Math. 16 (1975), 197–220.
M. A. Olshanetsky and A. M. Perelomov, Classical integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), No. 5, 313–400.
M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), No. 6, 313–404.
A. P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992), No. 5, 703–705.
B. S. Shastry and B. Sutherland, Solution of some integrable one-dimensional quantum systems, Phys. Rev. Lett. 71 (1993), No. 1, 5–8.
B. Sutherland, Exact results for a quantum many-body problem in one dimension, Phys. Rev. A 4 (1971), 2019–2021.
B. Sutherland, Exact results for a quantum many-body problem in one dimension. II, Phys. Rev. A 5 (1972), 1372–1376.
B. Sutherland, Exact ground-state wave function for a one-dimensional plasma, Phys. Rev. Lett. 34 (1975), 1083–1085.
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982), No. 2, 253–316.
S. Wojciechowski, An integrable marriage of the Euler equations with the Calogero-Moser system, Phys. Lett. A 111 (1985), No. 3, 101–103.
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Polychronakos, A.P. (2000). Generalizations of Calogero Systems. In: van Diejen, J.F., Vinet, L. (eds) Calogero—Moser— Sutherland Models. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1206-5_24
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DOI: https://doi.org/10.1007/978-1-4612-1206-5_24
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