论文标题
基本数值范围和Simon的定理在吸收特征值方面
The essential numerical range and a theorem of Simon on the absorption of eigenvalues
论文作者
论文摘要
令$ a（t）$为复杂的希尔伯特空间$ \ mathcal {h} $的（b）类型（b）的自动伴侣操作员的全态家族。 Kato-Rellich扰动理论说，只要它们保持在$ a（t）$的基本频谱的最低水平以下，$ a（t）$的孤立特征值将具有$ t $的分析功能。在阈值$ t_0 $中，这些特征值函数之一达到了基本频谱，基本频谱中的相应点可能或可能不是$ a（t_0）$的特征值。我们的结果将Simon定理推广到基本频谱的最小值为$ a（t_0）$的特征值提供足够的条件，这是根据特征值接近必需频谱的速率。我们还表明，$ a（t）$的特征值可以从下面接近基本频谱的速率对应于有限的自我接合操作员的特征值。这些结果背后的关键见解是基本的数值范围，该范围最近扩展到Bögli，Marletta和Tretter的无界运营商。
Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essential spectrum to be an eigenvalue of $A(t_0)$ based on the rate at which eigenvalues approach the essential spectrum. We also show that the rates at which the eigenvalues of $A(t)$ can approach the essential spectrum from below correspond to eigenvalues of a bounded self-adjoint operator. The key insight behind these results is the essential numerical range which was recently extended to unbounded operators by Bögli, Marletta, and Tretter.
