Revisiting the Christ-Kiselev's multi-linear operator technique and its applications to Schr\"odinger operators

TL;DR

This paper extends the Christ-Kiselev multi-linear operator technique to analyze Schrödinger operators, leading to new spectral results including criteria for spectrum types and bounds on spectral dimensions.

Contribution

The paper introduces a generalized multi-linear operator technique tailored for Schrödinger operators, enabling new spectral analysis methods and results.

Findings

WKB solutions for perturbed periodic Schrödinger operators

Sharp criteria for preservation of absolutely continuous spectra

Bounds on Hausdorff dimension of singular spectra

Abstract

We established a generalized version of the Christ-Kiselev's multi-linear operator technique to deal with the spectral theory of Schr\"odinger operators. As applications, several spectral results of perturbed periodic Schr\"odinger operators are obtained, including WKB solutions, sharp transitions of preservation of absolutely continuous spectra, criteria of absence of singular spectra and sharp bounds of the Hausdorff dimension of singular spectra.