Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions


In this paper we consider various regularity results for discrete quasi-periodic Schrödinger equations
-\psi_{n+1} – \psi_{n-1} +V(\theta +n\omega)\psi_n E\psi_n
with analytic potential $V$. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Hölder continuous in the energy provided $\omega$ has a typical continued fraction expansion. The proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the “avalanche-principle”. The latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks. In the multi-frequency case the integrated density of states is shown to have a modulus of continuity of the form $\exp(-|\log t|^\sigma)$ for some $0\lt \sigma\lt 1$, but currently we do not obtain Hölder continuity in the case of more than one frequency. We also present a mechanism for proving the positivity of the Lyapunov exponent for large disorders for a general class of equations. The only requirement for this approach is some weak form of a large deviation theorem for the Lyapunov exponents. In particular, we obtain an independent proof of the Herman-Sorets-Spencer theorem in the multi-frequency case. The approach in this paper is related to the recent nonperturbative proof of Anderson localization in the quasi-periodic case by J. Bourgain and M. Goldstein.



Michael Goldstein

Wilhelm Schlag