Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
复制标题
大狄利克雷特征值的边界准正交性和锐包含界
DOI：
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发表时间：
2010
影响因子：
2.9
通讯作者：
Andrew Hassell
中科院分区：
文献类型：
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作者：
A. Barnett;Andrew Hassell
本文研究了具有分段光滑边界的有界区域$Omegasubsetmathbb {R}^n $上Dirichlet Laplacian的特征函数$phi_j $和特征值$E_j $。本文用Helmholtz方程（Delta + E）u = 0的正规化试探解u的边界L^2范数来界定任意参数E> 0与谱{Ej}之间的距离.我们还从一个特征函数中得到了这个试解的误差的L^2范数。这两个结果是尖锐的常数，持有所有$E $大于一个小常数，并改善后，最有名的边界的莫勒佩恩的一个因素的波数$sqrt {E}$。一个应用是在高频率的本征值问题的解决方案，通过，例如，特解的方法。在平面的情况下，严格的星形域，我们给一个包含界的常数也是尖锐的。我们给出了明确的常数定理，并显示了一个数值例子，在2500年左右的特征值计算到14位的相对精度。证明利用了一个新的准正交性质的边界正常的衍生物的本征模式（定理1.3），在其本身的权利感兴趣。也就是说，秩为1的算子之和$partial_nphi_jlanglepartial_nphi_j，cdot的算子范数
在宽度为$sqrt {E}$的谱窗中的所有$E_j $上的角$--约为$E ^{（n-1）/2}$项的和--至多是大于任何一项的算子范数的常数因子（与$E $无关）。
We study eigenfunctions $phi_j$ and eigenvalues $E_j$ of the Dirichlet Laplacian on a bounded domain $Omegasubsetmathbb{R}^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E>0$ and the spectrum ${E_j}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(Delta+E)u=0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler-Payne by a factor of the wavenumber $sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators $partial_nphi_jlanglepartial_nphi_j,cdot
angle$ over all $E_j$ in a spectral window of width $sqrt{E}$—a sum with about $E^{(n-1)/2}$ terms—is at most a constant factor (independent of $E$) larger than the operator norm of any one individual term.