Quantum Toda systems and Lax pairs
Using the \({\mathbb{P}}\) operator [cf. \textit{L. D. Faddeev}, Integrable models in \((1+1)\)-dimensional quantum field theory, in ``Développements Récents en Théorie des Champs et Mécanique Statistique/Recent Advances in Field Theory and Statistical Mechanics'', Les Houches, Session XXXIX, 2 Août-10 Septembre 1982, J.-B. Zuber and R. Stora, eds., North-Holland, Amsterdam, pp. 561-608 (1984)], the author first constructs Lax pairs \(\{\) A,B\(\}\) for quantized systems that reduce to classical pairs when \(\hslash \to 0.\) In Section 3 that general approach is applied to the quantum mechanical Toda lattice, use being made of the formula for \({\mathbb{P}}\) given by \textit{D. I. Olive} and \textit{N. Turok} [Algebraic structure of Toda systems, Nucl. Phys. B 220, 491-507 (1983); cf. pp. 498-499]. A constant term appearing in the expression for B reproduces a term introduced by \textit{P. Mansfield} [Solution of Toda systems, Nucl. Phys. B 208, 277-300 (1982); cf. p. 288], who followed the approach of A. N. Leznov and M. V. Savel'ev. Conserved quantities are obtained in Section 4 for every power of a gauge transformation of A. Hermiticity properties discussed in an appendix then yield a set of commuting operators. Finally, an application to the Toda lattice on SU(3) is made. Quantum Toda systems have also been considered by, e.g., \textit{M. C. Gutzwiller} [Ann. Phys. 133, 304-331 (1981)], and \textit{R. Goodman} and \textit{N. R. Wallach} [Commun. Math. Phys. 105, 473-509 (1986; Zbl 0616.22010)].
scientific article; zbMATH DE number 4034711
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