Based on Christ–Kiselev’s multi-linear operator techniques, we prove several spectral results of perturbed periodic Schrdinger operators, including WKB type solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra, and sharp bounds of the Hausdorff dimension of singular continuous spectra.

中文翻译：

---

重温 Christ-Kiselev 的多线性算子技术及其在 Schrdinger 算子中的应用

基于 Christ-Kiselev 的多线性算子技术，我们证明了扰动周期 Schrdinger 算子的几个谱结果，包括 WKB 类型的解、绝对连续谱保存的锐跃迁、奇异谱不存在的标准和 Hausdorff 维的锐界奇异连续谱。