Abstract
It is shown that the inverse scattering method applies to both the classical and the quantum Goryachev-Chaplygin top. A new method, based on theR-matrix formalism, is proposed for deriving the equations determining the spectrum of the quantum integrals of motion. This method is of a rather general nature and may serve as an alternative to the so-called algebraic Bethe Ansatz.
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Literature cited
G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems of the Dynamics of Rigid Body [in Russian], Kiev (1978).
S. A. Chaplygin, “A new case of rotation of a rigid body propped up at one point,” in: Collected Works [in Russian], Vol. 1, Moscow (1948), pp. 118–124.
L. N. Sretenskii, “On some cases of motion of a heavy rigid body with gyroscope,” Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 3, 60–71 (1963).
I. V. Komarov, “The Goryachev-Chaplygin top in quantum mechanics,” Teor. Mat. Fiz.,50, No. 3, 402–409 (1982).
I. V. Komarov and V. V. Zalipaev, “The Goryachov-Chaplygin gyrostat in quantum mechanics,” J. Phys. A,17, 31–49 (1984).
L. D. Faddeev, “Quantum completely integrable models of field theory,” in: Proceedings of the 5th International Meeting on Nonlocal Field Theories [in Russian], Alushta, 1979, Dubna (1979), pp. 249–299.
P. P. Kulish and E. K. Sklyanin, “The quantum spectral transform method,” in: Lecture Notes in Phys., Vol. 151, Springer-Verlag, Berlin (1982), pp. 61–119.
M. S. Gutzwiller, “The quantum-mechanical Toda lattice, II,” Ann. Phys.,133, No. 2, 304–331 (1981).
H. Flaschka and D. W. McLaughlin, “Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions,” Progr. Theor. Phys.,55, No. 2, 438–456 (1976).
M. Kac and P. Van Moerbeke, “A complete solution of the periodic Toda problem,” Proc. Nat. Acad. Sci. USA,72, No. 8, 2879–2880 (1975).
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon (1977).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 236–257, 1984.
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Sklyanin, E.K. Goryachev-Chaplygin top and the inverse scattering method. J Math Sci 31, 3417–3431 (1985). https://doi.org/10.1007/BF02107243
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DOI: https://doi.org/10.1007/BF02107243