Schrodinger operators with fairly arbitrary spectral features

It is shown, using methods of inverse-spectral theory, that there exist Schrodinger operators on the line with fairly general spectral features. Thus, for instance, it follows from the main theorem, that if Sigma is any perfect subset of (-infinity,0], then there exist potentials q(j),j = 1,2 such that the associated Schrodinger operators H-j are self-adjoint and satisfy: sigma(H-j) = Sigma boolean OR, [0, infinity), sigma(ac)(H-j) [0, infinity), sigma(pp)(H-1) = sigma(sc)(H-2) = Sigma. The main result also implies the existence of states with interesting transport properties.