Whittaker function

Abstract: Automorphic forms on GL(2).- The differential equations satisfied by Whittaker functions.- Jacquet's Whittaker functions.- Fourier expansions of automorphic forms.- Invariants of G?\G.- Ramanujan sums on GL(3).- Eisenstein series.- The analytic continuation and functional equations satisfied by the L-series associated with an automorphic form.- Hecke operators and L-series.- The Mellin transforms of the Whittaker functions.

Abstract: Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for 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 N=2,3 and 4-particle chain. Our method to solve the problem combines the ideas of Gutzwiller and R-matrix approach of Sklyanin with the classical results in the theory of the 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 the natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.