Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$

Abstract

We prove the Hölder continuity of the integrated density of states for a class of quasi-periodic long-range operators on ${\ell }^2({\mathbb{Z}}^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the Hölder exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag \cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,\C)$ cocycles which is proved by quantitative almost reducibility method.

References in corpus (6)

• A Nonperturbative Eliasson's Reducibility Theorem
• Complex one-frequency cocycles
• Almost reducibility and absolute continuity I
• Continuity of the Lyapunov exponents for quasiperiodic cocycles
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Arithmetic version of Anderson localization via reducibility

Cited by in corpus (1)

• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles