可积本征函数的分布律
Distribution laws for integrable eigenfunctions
摘要：
我们确定了环面作用在环面Kahler簇上的联合本征函数的渐近性。这类簇是复几何中完全可积系统的模型。我们首先确定的特征函数的逐点渐近，这表明他们的行为像高斯集中在相应的经典环面。然后，我们表明，有一个普遍的高斯标度极限的分布函数在其中心附近。我们还确定了大尺度上本征函数尾部的极限分布。这些都不是普遍的，但取决于全球几何的复曲面品种，特别是对细节的指数衰减的本征函数远离经典允许的设置。
英文摘要：
We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.
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