The absolutely continuous spectrum of one-dimensional Schr"odinger operators

TL;DR

This paper investigates the structural properties of one-dimensional Schrödinger operators with absolutely continuous spectrum, revealing that their omega limit points are reflectionless on the spectral support, leading to new theorems about potential behavior.

Contribution

It extends the understanding of spectral properties of Schrödinger operators by establishing reflectionless conditions for omega limit points and deriving related theorems, including an Oracle Theorem.

Findings

Omega limit points are reflectionless on the spectral support.

Establishment of an Oracle Theorem for such potentials.

Extension of results from discrete to continuous Schrödinger operators.

Abstract

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.