Critical sets of random linear combinations of eigenfunctions
Critical sets of random linear combinations of eigenfunctions
复制标题
特征函数随机线性组合的临界集
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
L. Nicolaescu
中科院分区：
文献类型：
--
作者：
L. Nicolaescu
给定一个m维无边界（M，g）的紧致连通Riemann流形和一个大的正常数L，我们用UL表示C∞（M）中对应于特征值≤ L的Laplacian的特征函数所张成的子空间.我们用UL上的L-度量导出的标准高斯概率测度来装备UL，并且我们用NL表示UL中随机函数的临界点的期望数目。我们证明当L → ∞时NL <$Cm dimUL，其中Cm是一个显式的正常数，它只依赖于M的维数m，并且当m→∞时满足渐近估计log Cm <$m2 log m。内容
Given a compact, connected Riemann manifold without boundary (M, g) of dimension m and a large positive constant L we denote byUL the subspace of C∞(M) spanned by eigenfunctions of the Laplacian corresponding to eigenvalues≤ L. We equipUL with the standard Gaussian probability measure induced by the L-metric onUL, and we denote by NL the expected number of critical points of a random function in UL. We prove that NL ∼ Cm dimUL as L → ∞, where Cm is an explicit positive constant that depends only on the dimension m of M and satisfying the asymptotic estimate log Cm ∼ m2 log m as m→∞. CONTENTS