论文标题
Schrödinger运营商的光谱属性，具有本地$ H^{ - 1} $电势
Spectral properties of Schrödinger operators with locally $H^{-1}$ potentials
论文作者
论文摘要
我们使用本地$ h^{ - 1} $势头学习半行Schrödinger运营商。在第一部分中，我们关注此类操作员的一般光谱理论框架，包括对绝对连续光谱的最后一个类型的描述和不同光谱类型的足够条件。在第二部分中，我们专注于在本地$ h^{ - 1} $ sense中腐烂的潜力；我们在短距离和远程电位之间建立了光谱过渡，以及稀疏单数电位的$ \ ell^2 $光谱过渡。用于处理分布电位的正规化程序也非常适合控制电位的快速振荡。因此，即使在平稳的电势中，我们的结果也适用于通常不会被认为是腐烂甚至有界限的情况。
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 $\ell^2$ 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.
