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Integral representation for the eigenfunctions of a quantum periodic Toda chain. (English) Zbl 0970.37056
Authors’ summary: An integral representation for the eigenfunctions of a quantum periodic Toda chain is constructed for the $$N$$-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for the $$N=2,3$$ and $$4$$-particle chain. Our method of solving the problem combines the ideas of Gutzwiller and the $$R$$-matrix approach of Sklyanin with the classical results in the theory of Whittaker functions. In particular, we calculate Sklyanin’s invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining a natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
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