Generalized Fourier transform for the Camassa-Holm hierarchy

The squared eigenfunctions of the spectral problem associated with the Camassa-Holm equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the Camassa-Holm hierarchy as a generalized Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions.