The one-dimensional Schroedinger equation and a periodic potential

After recalling basic facts from the Titchmarsh-Weyl theory the linear matrix equation was derived and investigated which holds for functions related to the spectral matrix of the one-dimensional periodic Schroedinger equation. The Weyl solutions of the Schroedinger equation are used in the solution of this equation and associated nonlinear equations of the Milne type. Two distinct trace formulae reconstructing the potential simply follow from the transformed and modified Milne equations. The necessary ''spectral data'' of the inverse problem are determined by an infinite system of nonlinear first-order ordinary differential equations. Non-uniqueness of the solution of the inverse problem is confirmed by writing a broad variety of the isospectral Darboux transformations. (author). 12 refs