Baxter Q-operator and Givental integral representation for C_n and D_n

Abstract

Recently integral representations for the eigenfunctions of quadratic open Toda chain Hamiltonians for classical groups was proposed. This representation generalizes Givental representation for A_n. In this note we verify that the wave functions defined by these integral representations are common eigenfunctions for the complete set of open Toda chain Hamiltonians. We consider the zero eigenvalue wave functions for classical groups C_n and D_n thus completing the generalization of the Givental construction in these cases. The construction is based on a recursive procedure and uses the formalism of Baxter Q-operators. We also verify that the integral Q-operators for C_n, D_n and twisted affine algebra A_{2n-1}^{(2)} proposed previously intertwine complete sets of Hamiltonian operators. Finally we provide integral representations of the eigenfunctions of the quadratic $D_n$ Toda chain Hamiltonians for generic nonzero eigenvalues.