On the Kotani-Last Conjecture for the Dirac Operator

2025/09/23 by Nyah Davis, Davis, Nyah, Íris Emilsdóttir +4
Mathematics · #Spectral Theory in Mathematical Physics #Holomorphic and Operator Theory #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.2509.19072

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 Schrödinger operators. In particular, we disprove the Kotani--Last conjecture in the setting of one-dimensional Dirac operators.

Citations

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Related

• On the Kotani-Last and Schrodinger conjectures
• Limit-Periodic Dirac Operators with Thin Spectra
• The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators
• Absolutely continuous spectrum for limit-periodic Schrödinger operators
• The absolutely continuous spectrum of the almost Mathieu operator