Spectral properties of Schrödinger operators with locally H⁻¹ potentials

We study half-line Schrödinger operators with locally H⁻¹ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last–Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local H⁻¹ sense; we establish a spectral transition between short-range and long-range potentials and an ² spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded.