标题： 随机薛定谔方程的弱耦合极限：平均波函数 显示英文标题

标题： The weak coupling limit for the random Schrödinger equation: The average wave function

摘要： 我们考虑具有时间不变的弱随机势的薛定谔方程，其强度为$\epsilon\ll 1$，具有高斯统计特性。 我们证明当初始条件的变化尺度远大于势的关联长度时，在时间尺度$t\sim\epsilon^{-2}$上，补偿后的波函数收敛到一个确定性的极限。 这是在尖锐的假设下得出的，即随机势的相关函数$R(x)$的衰减比$1/|x|^2$更慢，这确保了有效势是有限的。 当$R(x)$的衰减比$1/|x|^2$更慢时，只要初始条件是“足够宏观的”，我们在比$\epsilon^{-2}$更短的时间尺度上建立了平均波函数的异常扩散行为。 我们还考虑了当初始条件与随机势在同一尺度上变化的动能区域，并得到了相关函数衰减快于$1/|x|^2$的势的平均波函数的极限。我们使用了 Schonberg 类的随机势，这使我们能够避开振荡相位估计。 显示英文摘要

摘要： We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale $t\sim\epsilon^{-2}$. This is shown under the sharp assumption that the correlation function $R(x)$ of the random potential decays slower than $1/|x|^2$, which ensures that the effective potential is finite. When $R(x)$ decays slower than $1/|x|^2$ we establish an anomalous diffusive behavior for the averaged wave function on a time scale shorter than $\epsilon^{-2}$, as long as the initial condition is "sufficiently macroscopic". We also consider the kinetic regime when the initial condition varies on the same scale as the random potential and obtain the limit of the averaged wave function for potentials with the correlation functions decaying faster than $1/|x|^2$. We use random potentials of the Schonberg class which allows us to bypass the oscillatory phase estimates.