Weyl's theory studies and the one-dimensional almost periodic Schroedinger equation

Making use of the Weyl's basis, we derive the modified Milne equation associated with the Schroedinger equation and find it's almost periodic solution, by means of which the Bloch-Weyl solutions can be written. There exist, in general, the generalized Bloch-Weyl functions only within spectral (allowed) bands of the energy parameter and on the negative energy half-axis linking up with the lowest allowed band. One solves the inverse problem in two ways each of them giving distinct formulas for reconstructing a potential. The Darboux transformation proves isospectral. (orig.)