Spectral Enclosure and Superconvergence for Eigenvalues in Gaps
Spectral Enclosure and Superconvergence for Eigenvalues in Gaps
复制标题
间隙特征值的谱域和超收敛
DOI：
10.1007/s00020-015-2247-0
复制
发表时间：
2014
影响因子：
0.8
通讯作者：
Michael Strauß
中科院分区：
文献类型：
--
作者：
J. Hinchcliffe;Michael Strauß
考虑了当直接应用有限截面伽辽金方法不可靠时，如何计算自伴随算子的特征值问题。在过去的二十年里，我们看到了所谓的二次方法的发展，以解决这个问题。最近出现了一种新的摄动方法，其思想是使特征值偏离实线，从而远离伽辽金方法失效的区域。我们提出了一种简化的摄动方法，它不需要<s:1>先验信息，并对其进行了严格的收敛分析。后者表明，在一般情况下，我们的方法将显著优于二次方法。我们也给出了形式为a + iB的算子的一个新的谱框，其中a是自伴随的，B是自伴随的且有界的。这使我们能够非常精确地控制特征值如何受到实线的扰动。用磁流体力学、Schrödinger和狄拉克算符等实例对主要结果进行了论证。
We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no á priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.