Holder continuity of the integrated density of states for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions
Holder continuity of the integrated density of states for quasi-periodic Schrodinger equations and averages of shifts of subharmonic functions
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DOI：
10.2307/3062114
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发表时间：
2001-07
影响因子：
4.9
通讯作者：
Michael Goldstein;W. Schlag
中科院分区：
文献类型：
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作者：
Michael Goldstein;W. Schlag
在本文中，我们考虑了离散准周期薛定谔方程 --n+l - Pn-1 + V(9 + nw)on = EOn 的各种正则性结果，解析势为 V。我们证明，在 Lyapunov 指数的正值区间上，只要 w 具有典型的连分数展开式，在能量上积分态密度是 Holder 连续的。证明基于某些单向矩阵范数的锐大偏差定理和“雪崩原理”。后者指的是一种机制，允许我们将单性矩阵的范数写为许多短块的范数的乘积。在多频率情况下，积分态密度在 0 < a < 1 的情况下具有 exp(- log tl) 形式的连续性模，但目前我们在多于一个频率的情况下无法获得 Holder 连续性。我们还提出了一种机制，用于证明一般方程组的大无序的李雅普诺夫指数的正性。这种方法的唯一要求是李亚普诺夫指数的大偏差定理的某种弱形式。特别是，我们在多频率情况下获得了 Herman-Sorets-Spencer 定理的独立证明。本文中的方法与 J. Bourgain 和 M. Goldstein 最近在准周期情况下对安德森定域性的非微扰证明有关。
In this paper we consider various regularity results for discrete quasiperiodic Schr6dinger equations --n+l - Pn-1 + V(9 + nw)on = EOn with analytic potential V. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Holder continuous in the energy provided w 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 tl) for some 0 < a < 1, but currently we do not obtain Holder 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.