Spectral properties of Schr\"odinger operators with locally $H^{-1}$ potentials

TL;DR

This paper investigates the spectral characteristics of Schr"odinger operators with locally $H^{-1}$ potentials, providing a comprehensive framework and identifying spectral transitions for decaying and sparse singular potentials.

Contribution

It introduces a spectral theoretic framework for such operators, including a Last--Simon-type description and conditions for spectral types, and explores spectral transitions for decaying and sparse potentials.

Findings

Spectral transition between short-range and long-range potentials.

$ ext{l}^2$ spectral transition for sparse singular potentials.

Applicable to smooth potentials with rapid oscillations.

Abstract

We study half-line Schr\"odinger 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 $\ell^2$ 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.