含随机杂质环面上的均匀分布本征函数
Uniformly distributed eigenfunctions on tori with random impurities
摘要：
本文研究了一类随机Schroedinger算子--具有N个独立均匀分布的随机δ势的Laplacian算子T^d_L = R^d/LZ^d，d = 2，3，其中L > 0是大的.我们根据环面的尺寸L、势的密度ρ= N/L^d和本征函数E的能量确定一个条件，使得任何这样的本征函数将以非零概率均匀分布在整个环面上。我们注意到，我们在这里证明的均匀分布仍然是符合本地化制度，本地化长度远大于尺寸的环面。事实上，我们的结果意味着一定的多项式下界的本地化长度，所以本地化长度变得无限大，因为E趋于无穷大。
英文摘要：
We study a random Schroedinger operator, the Laplacian with N independently uniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d = 2, 3, where L > 0 is large. We determine a condition in terms of the size of the torus L, the density of the potentials ρ= N/L^d and the energy of the eigenfunction E such any such eigenfunctions will with nonzero probability be uniformly distributed on the entire torus. We remark that the equidistribution we prove here is still consistent with a localized regime, where the localization length is much larger than the size of the torus. In fact our result implies a certain polynomial lower bound on the localization length, so the localization length becomes infinitely large as E tends to infinity.
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