The absolutely continuous spectrum of Jacobi matrices

TL;DR

This paper investigates the properties and implications of the absolutely continuous spectrum in one-dimensional Schrödinger operators, including new theorems and results on limit potentials and spectral predictions.

Contribution

It introduces an Oracle Theorem for potential prediction and proves a Denisov-Rakhmanov type theorem for finite gap cases, advancing understanding of spectral properties.

Findings

Established an Oracle Theorem for potential prediction.

Proved a Denisov-Rakhmanov type theorem for finite gap cases.

Demonstrated the difficulty of producing absolutely continuous spectrum in one dimension.

Abstract

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.