Schrödinger Operators with Fairly Arbitrary Spectral Features

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