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SIGMA 2 (2006), 035, 11 pages hep-th/0603140
https://doi.org/10.3842/SIGMA.2006.035
Calogero Model(s) and Deformed Oscillators
Marijan Milekovic a, Stjepan Meljanac b and Andjelo Samsarov b
a) Physics Department, Faculty of Science, Bijenicka c. 32, 10002 Zagreb, Croatia
b) Rudjer Boskovic Institute, Bijenicka c. 54, 10002 Zagreb, Croatia
Received November 30, 2005, in final form March 02, 2006; Published online March 17, 2006
Abstract
We briefly review some recent results concerning
algebraical (oscillator) aspects of the N-body single-species
and multispecies Calogero models in one dimension. We show how
these models emerge from the matrix generalization of the
harmonic oscillator Hamiltonian. We make some comments on the
solvability of these models.
Key words:
Calogero model; deformed oscillator algebra; SN-extended Heisenberg algebra.
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References
- Mattis D.C., The many-body problem:
70 years of exactly solved quantum many-body problems, Singapore,
World Scientific, 1993.
- Sutherland B., Beautiful models, Singapore, World Scientific, 2004.
- Calogero F., Ground state of one-dimensional N-body system, J. Math. Phys., 1969,
V.10, 2197-2200.
- Calogero F., Solution of the one-dimensional N-body
problems with quadratic and/or inversly quadratic pair potentials,
J. Math. Phys., 1971, V.12, 419-436.
- Van Dijen J.F., Vinet L. (Editors), Calogero-Moser-Sutherland models,
CRM Series in Mathematical Physics, Vol. 25, Berlin,
Springer, 2000.
- Kawakami N., Critical properties of quantum many-body
systems with 1/r2 interaction, cond-mat/9402011.
- Gibbons G.W., Townsend P.K., Black holes and Calogero models, Phys.
Lett. B, 1999, V.454, 187-192, hep-th/9812034.
- Sutherland B., Quantum many-body problem in one dimension: ground state,
J. Math. Phys., 1971, V.12, 246-250.
- Gomis J., Kapustin A., Two-dimensional unoriented strings and matrix models,
JHEP, 2004, V.0406, 002, 38 pages, hep-th/0310195.
- Gurappa N., Panigrahi P.P.,
Equivalence of the Calogero-Sutherland model to free harmonic
oscillators, Phys. Rev. B, 1999, V.59, R2490-R2493,
cond-mat/9710035.
- Ezung M., Gurappa N., Khare A., Panigrahi P.K.,
Quantum many-body systems with nearest and next-to-nearest
neighbor long-range interaction, Phys. Rev. B, 2005, V.71,
125121, 8 pages, cond-mat/0007005.
- Sutherland B., Exact results for a quantum many-body problem in one dimension II,
Phys. Rev. A, 1972, V.5, 1372-1376.
- Hallnas M., Langmann E., Explicit formulas for the eigenfunctios of
the N-body Calogero model, math-ph/0511040.
- Brink L., Hansson T.H., Vasiliev M.A., Explicit solution to the N-body Calogero problem,
Phys. Lett. B, 1992, V.286, 109-111, hep-th/9206049.
- Meljanac S., Milekovic M., Samsarov A., A multispecies Calogero model, Phys. Lett. B,
2003, V.573, 202-208, hep-th/0307126.
- Meljanac S., Milekovic M., Samsarov A., Stojic M., Interacting families
of Calogero-type particles and SU(1,1) algebra, Modern Phys.
Lett. B, 2004, V.18, 603-612, hep-th/0405132.
- Meljanac S., Milekovic M., Samsarov A.,
Generalized Calogero model in arbitrary dimensions, Phys.
Lett. B, 2004, V.594, 241-246, hep-th/0405131.
- Haldane F.D.M., "Fractional statistics" in arbitrary dimensions:
a generalization of the Pauli principle, Phys. Rev. Lett.,
1991, V.67, 937-940.
- Bernard D., Wu Y.-S., A note on statistical interactions
and the thermodynamical Bethe ansatz, in New Developments of
Integrable Systems and Long-Ranged interaction Models, Editors
M.-L. Ge and Y.-S. Wu, Nankai Lectures on Mathematical Physics,
1995, 10-21, cond-mat/9404025.
- Murthy M.V.N., Shankar R., Thermodynamics of
a one-dimensional ideal gas with fractional exclusion statistics, Phys. Rev. Lett., 1994, V.73,
3331-3334, cond-mat/9404096.
- Furtlehner C., Ouvry S., Calogero model for distinguishable particles, Modern Phys. Lett. A,
1995, V.9, 503-509, hep-th/9407107.
- Sen D., A multispecies Calogero-Sutherland model, Nucl. Phys. B, 1996, V.479, 554-574.
- Mashkevic S., Towards a quantum-mechanical model for multispecies exclusion statistics,
Phys. Lett. A, 1997, V.233, 30-36.
- Guhr T., Kohler H., Supersymmetry and models for two kinds of interacting particles, Phys. Rev. E,
2005, V.71, 045102, 4 pages, math-ph/0408033.
- Meljanac S., Samsarov A., Matrix oscillator and Calogero-type models, Phys. Lett. B, 2004,
V.600, 179-184, hep-th/0408241.
- Bardek V., Jonke L., Meljanac S., Milekovic M.,
Calogero model, deformed oscillators and the collapse, Phys.
Lett. B, 2002, V.531, 311-315, hep-th/0107053.
- Meljanac S., Milekovic M., Stojic M.,
Permutation invariant algebras, a Fock space realization and the
Calogero model, Eur. Phys. J. C Part. Fields, 2002, V.24,
331-343, math-ph/0201061.
- Polychronakos A.P., Exchange operator formalism for integrable system of particles,
Phys. Rev. Lett., 1992, V.69, 703-705, hep-th/9202057.
- Jonke L., Meljanac S., Bosonic realization of algebras in the Calogero model, Phys. Lett. B,
2002, V.256, 149-156, hep-th/0106135.
- Jonke L., Meljanac S., Dynamical symmetry algebra of the Calogero model, Phys. Lett. B, 2001,
V.511, 276-284, hep-th/0105043.
- Meljanac S., Milekovic M., A unified view of multimode algebras with Fock-like representations,
Internat. J. Modern Phys. A, 1996, V.11, 1391-1412.
- Gorsky A., Nekrasov N., Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory,
Nucl. Phys. B, 1994, V.414, 213-238, hep-th/9304047.
- Calogero F., Marchioro C., Exact bound states of some N-body systems
with two- and three-body forces, J. Math. Phys., 1973, V.14,
182-184.
- Forrester P.J., Some multidimensional integrals related to many-body systems with the 1/r2
potential, J. Phys. A: Math. Gen., 1992, V.25, L607-L614.
- Dasniers de Veigy A., On the solution of the Calogero model
distinguishable particles in the domain of intermediate
statistics, Nucl. Phys. B, 1997, V.483, 580-600.
- Meljanac S., Samsarov A., Universal properties of
conformal quantum many-body systems, Phys. Lett. B, 2005,
V.613, 221-225, hep-th/0503174.
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