Schrödinger operators with fairly arbitrary spectral features

Abstract

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 Σ is any perfect subset of (-∞, 0], 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(H₁)=σ_sc(H₂) =Σ. The main result also implies the existence of states with interesting transport properties.