二阶谱的极限集合
Limiting set of second order spectra
AI总结：
本文严格研究二阶相对谱的极限，证明在温和条件下其一致极限包含自伴算子的有限重孤立特征值，为无谱污染的特征值数值逼近提供理论依据。
摘要：
M.Levitin 和 E.Shargorodsky 在近期文章 [math.SP/0212087] 中提出了使用所谓的“二阶相对谱”来寻找自伴算子在本质谱间隙中的特征值。设 $M$ 是作用在希尔伯特空间 $H$ 上的自伴算子。复数 $z$ 属于 $M$ 相对于有限维子空间 $\Lambda\subset dom M^2$ 的二阶谱，当且仅当 $(M-z)^2$ 到 $\Lambda$ 的截断不可逆。值得注意的是，这些集合似乎提供了一种估计特征值的通用方法，避免了大多数线性方法中存在的谱污染问题。在本文中，我们严格研究了使用二阶谱寻找特征值的各个方面。我们的主要结果表明，在关于 $M$ 的某些相当温和的假设下，当 $\Lambda$ 向 $H$ 增大时，二阶谱的一致极限包含 $M$ 的有限重数的孤立特征值。在应用中，本质谱可以解析计算，而正是这些特征值需要数值近似。因此，该方法似乎在很高的普遍性水平上结合了无污染和近似性。
英文摘要：
M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let $M$ be a self-adjoint operator acting on the Hilbert space $H$. A complex number $z$ is in the second order spectrum of $M$ relative to a finite dimensional subspace $Λ\subset dom M^2$ if and only if the truncation to $Λ$ of $(M-z)^2$ is not invertible. It is remarkable that these sets seem to provide a general method to estimating eigenvalues free from the problems of spectral pollution present in most linear methods. In this notes we investigate rigorously various aspects of the use of second order spectrum to finding eigenvalues. Our main result shows that, under certain fairly mild hypothesis on $M$, the uniform limit of the second order spectra, as $Λ$ increases toward $H$, contains the isolated eigenvalues of $M$ of finite multiplicity. In applications the essential spectrum can be computed analytically, while precisely these eigenvalues are the ones that should be approximated numerically. Hence this method seems to combine non-pollution and approximation at a very high level of generality.
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