An inverse scattering problem for a perturbed Hill's operator

We consider the inverse scattering problem for the operator $L=-d^2/dx^2+p(x)+q(x)$, $x\in R^1$. The perturbation potential $q$ is expressed in terms of the periodic potential $p$ and the scattering data. We also obtain identities for the eigenfunctions of the unperturbed Hill's operator $L_0=-d^2/dx^2+p(x)$.