Billiard本征函数的临界划分与节亏
Critical partitions and nodal deficiency of billiard eigenfunctions
摘要：
本文讨论了有界区域上具有Dirichlet边界条件的薛定谔算子的本征函数的节点域的个数。在一维中，第$n$特征函数具有$n$节点域。柯朗定理认为，在任何维度上，第$n$个特征函数的节点域的数目不能超过$n$。然而，在大于1的维度中，等式仅对1000个本征函数成立。因此，出现了“节点缺陷”。已知具有任意大指数$n$的本征函数的例子只有两个节点域。近年来，有人建议考虑域的分区，而不是本征函数。它表明在最近的一份文件由Helffer，霍夫曼-Ostenhof和Terracini，（在某些自然条件下）二分分区最大限度地减少最大的基态能量在子域的分区，对应于“柯朗夏普”本征函数，即那些与零节点亏损。在本文中，作者证明了，在某些一般性条件下，在二分均分中，节点均分恰好对应于一个类似泛函的临界点，节点亏等于在这一点上的Morse指数.这特别解释了为什么所有的最小划分都必须是Courant sharp。
英文摘要：
The paper addresses the the number of nodal domains for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains. It was suggested in the recent years to look at the partitions of the domain, rather than eigenfunctions. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) bipartite partitions minimizing the maximum of the ground-state energies in sub-domains of the partition, correspond to the "Courant sharp" eigenfunctions, i.e. to those with zero nodal deficiency. In this paper, the authors show, under some genericity conditions, among the bipartite equipartitions, the nodal ones correspond exactly to the critical points of an analogous functional, with the nodal deficiency being equal to the Morse index at this point. This explains, in particular, why all the minimal partitions must be Courant sharp.
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