Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model. (English) Zbl 1002.81012
Summary: Using a similarity transformation that maps the Calogero model into \(N\) decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the \(B_N\)-Calogero model.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81V70 | Many-body theory; quantum Hall effect |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
Keywords:
bosonic state; similarity transformation; decoupled quantum harmonic oscillators; simultaneous eigenfunctions; Jack polynomial; Dunkl operator formulationReferences:
| [1] | Sutherland, B., Exact results for a quantum many-body problem in one dimension, Phys. Rev. A, 4, 2019-2021 (1971) |
| [2] | Sutherland, B., Exact results for a quantum many-body problem in one dimension. II, Phys. Rev. A, 5, 1372-1376 (1972) |
| [3] | Jack, H., A class of symmetric polynomials with a parameter, Proc. R. Soc. Edinburgh Sect. A, 69, 1-18 (1970) · Zbl 0198.04606 |
| [4] | I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 2nd ed., 1995; I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 2nd ed., 1995 · Zbl 0824.05059 |
| [5] | Stanley, R. P., Some combinatorial properties of Jack symmetric functions, Adv. Math., 77, 76-115 (1989) · Zbl 0743.05072 |
| [6] | Forrester, P. J., Selberg correlation integrals and the \(1/r^2\) quantum many-body system, Nucl. Phys. B, 388, 671-699 (1992) |
| [7] | Forrester, P. J., Addendum to “Selberg correlation integrals and the \(1/r^2\) quantum many-body system”, Nucl. Phys. B, 416, 377-385 (1994) |
| [8] | Ha, Z. N.C., Exact dynamical correlation functions of Calogero-Sutherland model and one-dimensional fractional statistics, Phys. Rev. Lett., 73, 1574-1577 (1994) |
| [9] | Ha, Z. N.C., Fractional statistics in one-dimension: view from an exactly solvable model, Nucl. Phys. B, 435, 604-636 (1995) · Zbl 1020.82538 |
| [10] | Lapointe, L.; Vinet, L., A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture, IMRN, 9, 419-424 (1995) · Zbl 0868.33009 |
| [11] | Lapointe, L.; Vinet, L., Exact operator solution of the Calogero-Sutherland model, Commun. Math. Phys., 178, 425-452 (1996) · Zbl 0859.35103 |
| [12] | Calogero, F., Solution of the one-dimensional \(N\)-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12, 419-436 (1971) · Zbl 1002.70558 |
| [13] | Sutherland, B., Quantum many-body problem in one dimension: ground state, J. Math. Phys., 12, 246-250 (1971) |
| [14] | Olshanetsky, M. A.; Perelomov, A. M., Quantum integrable systems related to Lie algebras, Phys. Rep., 94, 313-404 (1983) |
| [15] | Perelomov, A. M., Algebraic approach to the solution of a one-dimensional model of \(N\) interacting particles, Theor. Math. Phys., 6, 263-282 (1971) |
| [16] | Brink, L.; Hansson, T. H.; Vasiliev, M. A., Explicit solution to the \(N\)-body Calogero problem, Phys. Lett. B, 286, 109-111 (1992) |
| [17] | Brink, L.; Hansson, T. H.; Konstein, S.; Vasiliev, M. A., The Calogero model – anyonic representation, fermionic extension and supersymmetry, Nucl. Phys. B, 401, 591-612 (1993) · Zbl 0941.81606 |
| [18] | Ujino, H.; Wadati, M., Algebraic approach to the eigenstates of the quantum Calogero model confined in an external harmonic well, Chaos, Solitons and Fractals, 5, 109-118 (1995) · Zbl 0824.58055 |
| [19] | Ujino, H.; Wadati, M., The quantum Calogero model and the \(W\)-algebra, J. Phys. Soc. Japan, 63, 3585-3597 (1994) · Zbl 0972.81695 |
| [20] | Ujino, H.; Wadati, M., Orthogonal symmetric polynomials associated with the quantum Calogero model, J. Phys. Soc. Japan, 64, 2703-2706 (1995) · Zbl 0972.81697 |
| [21] | Ujino, H.; Wadati, M., Algebraic construction of the eigenstates for the second conserved operator of the quantum Calogero model, J. Phys. Soc. Japan, 65, 653-657 (1996) · Zbl 0942.81621 |
| [22] | Ujino, H.; Wadati, M., Rodrigues formula for Hi-Jack symmetric polynomials associated with the quantum Calogero model, J. Phys. Soc. Japan, 65, 2423-2439 (1996) · Zbl 0948.39009 |
| [23] | Ujino, H.; Wadati, M., Orthogonality of the Hi-Jack polynomials associated with the Calogero model, J. Phys. Soc. Japan, 66, 345-350 (1997) · Zbl 0957.33010 |
| [24] | Baker, T. H.; Forrester, P. J., The Calogero-Sutherland model and generalized classical polynomials, Commun. Math. Phys.188, 175-216 (1997) · Zbl 0903.33010 |
| [25] | van Diejen, J. F., Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Commun. Math. Phys., 188, 467-497 (1997) · Zbl 0917.33008 |
| [26] | Kakei, S., An orthogonal basis for the \(B_N\)-type Calogero model, J. Phys. A: Math. Gen., 30, L535-L541 (1997) · Zbl 0933.33013 |
| [27] | Dunkl, C. F., Differential-difference operators associated to reflection group, Trans. Amer. Math. Soc., 311, 167-183 (1989) · Zbl 0652.33004 |
| [28] | Polychronakos, A. P., Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett., 69, 703-705 (1992) · Zbl 0968.37521 |
| [29] | Yamamoto, T., Multicomponent Calogero model of \(B_N\)-type confined in a harmonic well, Phys. Lett. A, 208, 293-302 (1995) · Zbl 1020.81971 |
| [30] | N. Gurappa, P.K. Panigrahi, Equivalence of Calogero-Sutherland model to free harmonic oscillators, preprint, cond-mat/9710035, 1997; N. Gurappa, P.K. Panigrahi, Equivalence of Calogero-Sutherland model to free harmonic oscillators, preprint, cond-mat/9710035, 1997 |
| [31] | N. Gurappa, P.K. Panigrahi, Mapping of the \(B_N\); N. Gurappa, P.K. Panigrahi, Mapping of the \(B_N\) |
| [32] | Sogo, K., A simple derivation of multivariable Hermite and Legendre polynomials, J. Phys. Soc. Japan, 65, 3097-3099 (1996) · Zbl 0958.33009 |
| [33] | Ujino, H.; Wadati, M., Algebraic construction of a new symmetric orthogonal basis for the Calogero model, J. Phys. Soc. Japan, 67, 1-4 (1998) · Zbl 0943.81055 |
| [34] | Lassalle, M., Polynômes de Laguerre généralisés, C. R. Acad. Sci. Paris t. Séries I, 312, 725-728 (1991) · Zbl 0739.33007 |
| [35] | Lassalle, M., Polynômes de Hermite généralisés, C. R. Acad. Sci. Paris t. Séries I, 313, 579-582 (1991) · Zbl 0748.33006 |
| [36] | L.I. Schiff, 1968 Quantum Mechanics, McGraw-Hill Book Company, New York, 3rd ed; L.I. Schiff, 1968 Quantum Mechanics, McGraw-Hill Book Company, New York, 3rd ed · Zbl 0068.40202 |
| [37] | Ujino, H.; Nishino, A.; Wadati, M., New non-symmetric orthogonal basis for the Calogero model with distinguishable particles, Phys. Lett. A, 249, 459-464 (1998) · Zbl 0941.81058 |
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