Counterexamples to the Kotani-Last Conjecture for Continuum Schr\"odinger Operators via Character-Automorphic Hardy Spaces

TL;DR

This paper constructs counterexamples to the Kotani-Last conjecture for continuum Schr"odinger operators by adapting methods involving character-automorphic Hardy spaces, revealing cases with absolutely continuous spectrum lacking almost periodicity.

Contribution

The paper introduces a novel approach to generate counterexamples for the Kotani-Last conjecture in continuum Schr"odinger operators using character-automorphic Hardy spaces and a new correspondence with Jacobi matrices.

Findings

Counterexamples with absolutely continuous spectrum and non-almost periodic coefficients.

Explicit correspondence between continuum Schr"odinger and Jacobi reflectionless operators.

Extension of the approach to extended CMV matrices.

Abstract

The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense in a variety of settings; for example, discrete Schr\"odinger operators, Jacobi matrices, CMV matrices, and continuum Schr\"odinger operators. In the main body of this paper we show how to construct counterexamples to the Kotani-Last conjecture for continuum Schr\"odinger operators by adapting the approach developed by Volberg and Yuditskii to construct counterexamples to the Kotani-Last conjecture for Jacobi matrices. This approach relates the reflectionless operators associated with the prescribed spectrum to a family of character-automorphic Hardy spaces and then relates the shift action on the space of operators to the resulting action on the associated characters. The key to our approach is an explicit correspondence between the space of continuum reflectionless Schr\"odinger operators associated with a given set and the space of reflectionless Jacobi matrices associated with a derived set. Once this correspondence is established we can rely to a large extent on the previous work of Volberg and Yuditskii to produce the resulting action on the space of characters. We analyze this action and identify situations where we can observe absolute continuity without almost periodicity. In the appendix we show how to implement this strategy and obtain analogous results for extended CMV matrices.