Avila's acceleration via zeros of determinants, and applications to Schrödinger cocycles

2022/12/12 by Rui Han, Wilhelm Schlag, Han, Rui +1
Mathematics · Physics and Astronomy · #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2212.05988

Abstract

In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove β-Hölder continuity of the integrated density of states for supercritical quasi-periodic Schrödinger operators restricted to the ℓ-th stratum, for any β<(2(ℓ-1))^-1 and ℓ≥2. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.

Related

• Hölder continuity of the integrated density of states for Liouville frequencies
• Sharp localization on the first supercritical stratum for Liouville frequencies
• Winding number, density of states and acceleration
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ℓ²(\Z^d)
• Absolute continuity of the integrated density of states in the localized regime