Anosov流形上Laplacian特征函数的半离域性
Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
摘要：
研究了具有Anosov测地线流的紧致黎曼流形上Laplacian的高能本征函数。与本征函数序列相关的半经典测度的局部化由该测度的柯尔莫哥洛夫-西奈熵来表征。我们表明，这个熵必然是有界的，从下面的一个常数，在恒定的负曲率的情况下，等于一半的最大熵。在这个意义上，高能本征函数至少是半离域的。
英文摘要：
We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.
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