## Complete integrability of relativistic Calogero-Moser systems and elliptic function identities.(English)Zbl 0673.58024

We recall that the potential energy of the (classical, nonrelativistic) Calogero-Moser system on the line is expressed in terms of the Weierstrass function $${\mathfrak p}(x)$$, degenerate cases of which are $$x^{-2}$$, $$\sin^{-2}x$$ and $$sh^{-2}x$$. In a previous paper with H. Schneider [Ann. Phys. 170, 370-405 (1986; Zbl 0608.35071)] the author exhibited sufficiently many independent constants of the motion in involution for Poincaré-invariant generalizations of those systems. In the present paper a formal quantization of those classical integrals is considered. The definition of the associated operators involves exponentials of the quantized rapidity variables and a meromorphic function that must satisfy a set of functional equations. In an appendix it is proved that a suitable quotient of $$\sigma$$-functions satisfies that set, and the vanishing of the quantum commutators is obtained as a consequence. A corollary shows that a translate of the $${\mathfrak p}$$- function satisfies a set of functional equations obtaind as a limit of the previous set, and the vanishing of the Poisson brackets of the classical integrals follows. Thus, for these models quantum integrability implies classical integrability.
This paper includes identities for the $$\sigma$$-function and generalized Cauchy determinantal identities that would seem to be of independent interest. The latter are used in the author’s extension to the elliptic case of the results obtained in op.cit. for the hyperbolic Lax matrix.
The author has kindly informed the reviewer that item [10] in the bibliography has already appeared; cf. the author in Commun. Math. Phys. 115, No.1, 127-165 (1988; Zbl 0667.58016)].
Reviewer: H.H.Torriani

### MSC:

 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 81S99 General quantum mechanics and problems of quantization 33E05 Elliptic functions and integrals 15A15 Determinants, permanents, traces, other special matrix functions

### Citations:

Zbl 0608.35071; Zbl 0667.58016
Full Text:

### References:

 [1] Ruijsenaars, S.N.M., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (N.Y.)170, 370-405 (1986) · Zbl 0608.35071 [2] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Reps.71, 313-400 (1981) [3] Ruijsenaars, S.N.M.: On one-dimensional integrable quantum systems with infinitely many degrees of freedom. Ann. Phys. (N.Y.)128, 335-362 (1980) [4] Ruijsenaars, S.N.M.: To appear [5] Erdélyi, A. (ed.): Higher transcendental functions, Vol. II. Florida: Krieger 1981 · Zbl 0542.33001 [6] Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1973 · JFM 45.0433.02 [7] Calogero, F., Marchioro, C., Ragnisco, O.: Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potentialV a (x)=g 2 a 2 sinh?2 ax. Lett. Nuovo Cimento13, 383-390 (1975) [8] Olshanetsky, M.A., Perelomov, A.M.: Quantum completely integrable systems connected with semi-simple Lie algebras. Lett. Math. Phys.2, 7-13 (1977) · Zbl 0366.58005 [9] Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Reps.94, 313-404 (1983) [10] Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems (to appear) · Zbl 0667.58016 [11] Krichever, I.M.: Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Funct. Anal. Appl.14, 282-290 (1980) · Zbl 0473.35071 [12] Calogero, F.: Explicitly solvable one-dimensional many-body problems. Lett. Nuovo Cimento13, 411-416 (1975) [13] Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties, and representation theory. Adv. Math.38, 318-379 (1980) · Zbl 0455.58010 [14] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601 [15] Carey, A.L., Hannabuss, K.C.: Temperature states on loop groups, theta-functions, and the Luttinger model. Canberra preprint, 1985 · Zbl 0633.46068 [16] Palmer, J., Tracy, C.: Two-dimensional Ising correlations: Convergence of the scaling limit. Adv. Appl. Math.2, 329-388 (1981) · Zbl 0474.46063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.