Complete Set of Eigenfunctions of the Quantum Toda Chain

The integral representation of the eigenfunctions of quantum periodic Toda chain constructed by Kharchev and Lebedev is revisited. We prove that Pasquier and Gaudin’s solutions of the Baxter equation provides a complete set of eigenfunctions under this integral representation. This will, in addition, show that the joint spectrum of commuting Hamiltonians of the quantum periodic Toda chain is simple.