On nonperturbative localization with quasi-periodic potential
On nonperturbative localization with quasi-periodic potential
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DOI：
10.2307/2661356
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发表时间：
2000-11
影响因子：
4.9
通讯作者：
J. Bourgain;Michael Goldstein
中科院分区：
文献类型：
--
作者：
J. Bourgain;Michael Goldstein
本文的两个主要结果涉及具有d频率的拟周期势的一维晶格薛定谔算子的安德森局域化。首先，在d = 1或2的情况下，证明了谱是具有指数衰减特征函数的纯点，对于所有势（用d维环面上的三角多项式定义），其中李亚普诺夫指数对于所有频率和所有能量都是严格正的。其次，对于每一个非常数实解析势和d频率的丢芬图集，给出了用一个足够大的常数重新标度的相同势的李亚普诺夫指数的下界。
The two main results of the article are concerned with Anderson Localization for one-dimensional lattice Schroedinger operators with quasi-periodic potentials with d frequencies. First, in the case d = 1 or 2, it is proved that the spectrum is pure-point with exponentially decaying eigenfunctions for all potentials (defined in terms of a trigonometric polynomial on the d-dimensional torus) for which the Lyapounov exponents are strictly positive for all frequencies and all energies. Second, for every non-constant real-analytic potential and with a Diophantine set of d frequencies, a lower bound is given for the Lyapounov exponents for the same potential rescaled by a sufficiently large constant.