On the Kotani-Last Conjecture for the Dirac Operator

Abstract

We prove a dichotomy of almost periodicity for reflectionless one-dimensional Dirac operators whose spectra satisfy certain geometric conditions, extending work of Volberg--Yuditskii. We also construct a weakly mixing Dirac operator with a non-constant continuous potential whose spectrum is purely absolutely continuous, adapting Avila's argument for continuous Schr\"odinger operators. In particular, we disprove the Kotani--Last conjecture in the setting of one-dimensional Dirac operators.