椭圆型伪微分算子随机结点集的贝蒂数
Betti numbers of random nodal sets of elliptic pseudo-differential operators
摘要：
给定一个下有界的椭圆自伴伪微分算子$P$，作用在光滑闭流形$M$上的黎曼线丛的截面上，当$L$增长到无穷大时，我们从上估计了取在特征值低于$L$的$P$的特征空间的直和中的随机截面的消失轨迹的贝蒂数.这些上估计遵循从一些均匀分布的临界点的限制，一个固定的莫尔斯函数，这个消失轨迹。然后我们考虑与$M$上的某些黎曼度量相关联的Laplace-Beltrami算子和Dirichlet-to-Neumann算子的例子。
英文摘要：
Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.
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