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References
P. D. Lax:Comm. Pure Appl. Math. 21, 467 (1968).
H. Flaschka:Phys. Rev. B,9, 1924 (1974);Prog. Theor. Phys.,51, 703 (1974).
S. V. Manakov:Žurn. Ėksp. Teor. Fiz.,67, 543 (1974).
J. Moser:Three integral Hamiltonian systems connected with iso-spectral deformations (preprint).
F. Calogero, C. Marchioro andO. Ragnisco:Lett. Nuovo Cimento,13, 383 (1975), hereafter referred to as CMR.
A. Erdélyi (Editor):Higher Transcendental Functions, Vol.3 (New York, N. Y., 1955).
This case was first solved in the quantal case: forN=3 byC. Marchioro:Journ. Math. Phys.,11, 2193 (1970), and for arbitraryN byF. Calogero:Journ. Math. Phys.,12, 419 (1971). In the classical case it was solved for arbitraryN byJ. Moser: ref. (4).J. Moser:Three integral Hamiltonian systems connected with iso-spectral deformations (preprint).
This case was first solved in the quantal case byB. Sutherland:Phys. Rev. A,5, 1375 (1972); then, in the classical case, byJ. Moser: ref. (4).Three integral Hamiltonian systems connected with iso-spectral deformations (preprint). The matrixL used byMoser corresponds to the solution α2(J).
B. Sutherland:Phys. Rev. A,8, 2514 (1973).
M. A. Olshanetzky andA. M. Perelomov:J. Phys. A (to be submitted to).
B. Sutherland: prprint ITP-SB-75-2 (to be published).
See, for instance, Subsect.13. 15 of ref. (8)A. Erdélyi (Editor):Higher Transcendental Functions, Vol.3 (New York, N. Y., 1955).
C. Marchioro: unpublished;D. C. Khandekar andS. V. Lawande:Amer. Journ. Phys.,40, 458 (1972).
See problem 38 inS. Flügge:Practical Quantum Mechanics, Vol.1 (Berlin, 1971).
See however, in this connection, ref.(22).
This opens the possibility to evaluate a number of nontrivial multiple integrals, followingG. Gallavotti andC. Marchioro:Journ. Math. Anal. Appl.,44, 661 (1973).
C. N. Yang (Phys. Rev. Lett. 19, 1312 (1967);Phys. Rev.,168, 1920 (1968) is also applicable in this case, at least for the special values of the coupling constantg 2=n(n−1), n=2,3, …, in which cases the two-body potential is transparent (vanishing reflection coefficient), but it hasn−1 bound states (and a zeroenergy resonance); in these cases also the group-theoretical structure underlying theN-body quantal problem is particularly transparent (13)M. A. Olshanetzky andA. M. Perelomov:J. Phys. A (to be submitted to).
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Calogero, F. Exactly solvable one-dimensional many-body problems. Lett. Nuovo Cimento 13, 411–416 (1975). https://doi.org/10.1007/BF02790495
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DOI: https://doi.org/10.1007/BF02790495