Uniform positivity of the Lyapunov exponent for the doubling map model with the Lipschitz-monotone potentials

Source abstract

This paper contributes to answering Damanik’s open problem [Damanik, Proc. Sympos. Pure Math. 76, 539 (2007)] by establishing a uniform lower bound of the Lyapunov exponent for Schrödinger operators with Lipschitz-monotone potentials generated by the doubling map. We generalize the work of Zhang [Zhang, Comm. Math. Phys. 405, 231 (2024)] by proving that C1-monotonicity can be relaxed to Lipschitz monotonicity. For a large coupling constant λ, we obtain the uniform lower bound L(E)≥log⁡λ−Cγ−−1, which highlights the critical influence of the Lipschitz constant γ−.