Localization of Laplacian Eigenfunctions in Simple and Irregular Domains
Localization of Laplacian Eigenfunctions in Simple and Irregular Domains
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简单域和不规则域中拉普拉斯特征函数的局部化
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
N. Thanh
中科院分区：
文献类型：
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作者：
N. Thanh
本论文的主要目的是研究有界区域中Laplacian特征函数的局部化问题，当一个特征函数主要由区域的一个小区域支撑并且在该区域外消失时。研究了简单区域和不规则区域中Dirichlet和Neumann边界条件下的高频和低频局部化问题。三种类型的高频本地化（回音壁，弹跳球，聚焦eigemodes）已被重新审视在圆形，球形和椭圆形域的特征函数的范数推导出明确的不等式。反过来，没有本地化已被发现在大多数矩形域，导致制定一个开放的问题，承认高频本地化的域的特征。利用Maslov型微分不等式，在不同的变截面分支区域中，低频Dirichlet本征函数的指数衰减得到了广泛的研究.在一个显式条件下，本征函数的L2-范数已被证明是沿着分支以显式计算的衰减率指数衰减。这个严格的上限，这是适用于任何尺寸和有限和无限的分支，提出了一个新的成就，在经典和量子波导理论，在微电子学，光学和声学的潜在应用。对于具有恒定截面轮廓的有界量子波导，导出了获得局域本征函数的分支长度的一个充分条件。本文证明了典型的有限量子波导（如L形、弯曲带和交叉带）在分支足够长的条件下存在陷波模，并对所需的最小长度作了精确的估计。本征模的局部化特征对分支长度和波导形状的高灵敏度可能潜在地用于微电子和光学中的开关器件。本文分析了一类平面谱图中局域本征模的性质。提出了一种求解无向赋权图的Laplacian矩阵特征值问题的分而治之算法，该算法比传统算法运行速度更快。已经开发了一种谱方法来研究反应介质中反射布朗运动的生存概率。生存概率已表示在拉普拉斯特征函数的谱分解的形式。反应区域的几何结构的作用及其对长时间制度中的总反应速率的影响进行了研究。该方法为设计高效催化剂或扩散交换器的最佳几何形状提供了数学基础。
The primary goal of the thesis is to study localization of Laplacian eigenfunctions in bounded domains when an eigenfunction is mainly supported by a small region of the domain and vanishing outside this region. The high-frequency and low-frequency localization in simple and irregular domains has been investigated for both Dirichlet and Neumann boundary conditions. Three types of high-frequency localization (whispering gallery, bouncing ball, and focusing eigemodes) have been revisited in circular, spherical and elliptical domains by deriving explicit inequalities on the norm of eigenfunctions. In turn, no localization has been found in most rectangular domains that led to formulating an open problem of characterization of domains that admit high-frequency localization. Using the Maslov-type differential inequalities, the exponential decay of low-frequency Dirichlet eigenfunctions has been extensively studied in various domains with branches of variable cross-sectional profiles. Under an explicit condition, the L2-norm of an eigenfunction has been shown to exponentially decay along the branch with an explicitly computed decay rate. This rigorous upper bound, which is applicable in any dimension and for both finite and infinite branches, presents a new achievement in the theory of classical and quantum waveguides, with potential applications in microelectronics, optics and acoustics. For bounded quantum waveguides with constant cross-sectional profiles, a sufficient condition on the branch lengths has been derived for getting a localized eigenfunction. The existence of trapped modes in typical finite quantum waveguides (e.g L-shape, bent strip and cross of two strips) has been proven provided that their branches are long enough, with an accurate estimate on the required minimal length. The high sensitivity of the localization character of eigenmodes to the length of branches and to the shape of the waveguide may potentially be used for switching devices in microelectronics and optics. The properties of localized eigenmodes in a class of planar spectral graphs have been analyzed. An efficient divide-and-conquer algorithm for solving the eigenproblem of the Laplacian matrix of undirected weighted graphs has been proposed and shown to run faster than traditional algorithms. A spectral approach has been developed to investigate the survival probability of reflected Brownian motion in reactive media. The survival probabilities have been represented in the form of a spectral decomposition over Laplacian eigenfunctions. The role of the geometrical structure of reactive regions and its influence on the overall reaction rate in the long-time regime has been studied. This approach presents a mathematical basis for designing optimal geometrical shapes of efficient catalysts or diffusive exchangers.