Abstract
The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its eigenfunctions can be expressed in terms of Jack symmetric polynomials. The exact solvability of the problem is explained by a hidden sl(3, ℝ) symmetry. A generalized Sutherland three-particle problem including both two- and three-body trigonometric potentials and internal degrees of freedom is then considered. It is analyzed in terms of three first-order noncommuting differential-difference operators, which are constructed by combining SUSYQM upercharges with the elements of the dihedral group D6 Three alternative commuting operators are also introduced.
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References
A. A. Andrianov, N. V. Borisov, M. I. Eides, and M. V. Ioffe, Super-symmetric origin of equivalent quantum systems, Phys. Lett. A 109 (1985), No. 4, 143–148.
A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, The factorization method and quantum systems with equivalent energy spectra, Phys. Lett. A 105 (1984), No. 1-2, 19–22.
D. Bernard, M. Gaudin, F. D. M. Haldane, and V. Pasquier, Yang-Baxter equation in long-range interacting systems, J. Phys. A 26 (1993), No. 20, 5219–5236, hep-th/9301084.
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 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), 1425–1430.
C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), No. 1, 167–183.
M. Hamermesh, Group Theory and Its Application to Physical Problems, Addison-Wesley Series in Physics, Addison-Wesley, Reading, MA, 1962.
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.
C. Quesne, Exchange operators and the extended Heisenberg algebra for the three-body Calogero-Marchioro-wolfes problem, Modern Phys. Lett. A10 (1995), No. 18, 1323–1330.
C. Quesne, Three-body generalization of the Sutherland model with internal degrees of freedom, Europhys. Lett. 35 (1996), 407–412.
C. Quesne, Exactly solvable three-particle problem with three-body interaction, Phys. Rev. A 55 (1997), No. 5, 3931–3934.
W. Rühl and A. Turbiner, Exact solvability of the Calogero and Sutherland models, Modern Phys. Lett. A10 (1995), No. 29, 2213–2221.
R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), No. 1, 76–115.
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.
J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974), 1420–1424.
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Quesne, C. (2000). Three-Body Generalizations of the Sutherland Problem. 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_25
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DOI: https://doi.org/10.1007/978-1-4612-1206-5_25
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