Quantum limits on flat tori
Quantum limits on flat tori
复制标题
平面环面的量子极限
DOI：
10.1090/s1079-6762-95-02004-x
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
D. Jakobson
中科院分区：
文献类型：
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作者：
D. Jakobson
我们对二维平面圆环上拉普拉斯归一化本征函数的平方的所有弱极限（称为量子极限）进行分类。我们还获得了关于三维及更高维度的此类限制的一些结果。许多结果都是几何引理的结果，该几何引理描述了 R^n 中余维一的单纯形的性质，其顶点是球面上的格点。该引理由两个佩尔方程组的解数有限性得出。引理的推论是 B. Connes 结果的概括。我们还指出了任何维度上平坦环面的量子极限的绝对连续性的证明（由 J. Bourgain 传达给我们）。将齐格蒙德的二维结果推广到三维之后，我们讨论该结果到更高维度的各种可能的推广以及与量子极限密度的 L^p 范数及其傅立叶级数的关系。
We classify all weak * limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (called quantum limits). We also obtain several results about such limits in dimensions three and higher. Many of the results are a consequence of a geometric lemma which describes a property of simplices of codimension one in R^n whose vertices are lattice points on spheres. The lemma follows from the finiteness of the number of solutions of a system of two Pell equations. A consequence of the lemma is a generalization of the result of B. Connes. We also indicate a proof (communicated to us by J. Bourgain) of the absolute continuity of the quantum limits on a flat torus in any dimension. After generalizing a two-dimensional result of Zygmund to three dimensions, we discuss various possible generalizations of that result to higher dimensions and the relation to L^p norms of densities of quantum limits and their Fourier series.