关于eigenvalue扰动理论中的一个问题
On a problem in eigenvalue perturbation theory
摘要：
考虑$K_t=K_0+tW$，$t\in [0,1]$型的加法扰动，其中$K_0$和$W$是可分Hilbert空间$\mathcal{H}$中的自伴算子，且$W$是有界的.另外，我们假设$W$的范围是生成的（即，循环）子空间。如果$λ_0$是$K_0$的特征值，则在$W$是非负的附加假设下，已知所有$t\in [0,1]$的集合的勒贝格测度为零，其中$λ_0$是$K_t$的特征值。我们回顾了这个结果及其证明，并通过明确的反例表明，非负性假设$W\geq 0$不能被删除。
英文摘要：
We consider additive perturbations of the type $K_t=K_0+tW$, $t\in [0,1]$, where $K_0$ and $W$ are self-adjoint operators in a separable Hilbert space $\mathcal{H}$ and $W$ is bounded. In addition, we assume that the range of $W$ is a generating (i.e., cyclic) subspace for $K_0$. If $λ_0$ is an eigenvalue of $K_0$, then under the additional assumption that $W$ is nonnegative, the Lebesgue measure of the set of all $t\in [0,1]$ for which $λ_0$ is an eigenvalue of $K_t$ is known to be zero. We recall this result with its proof and show by explicit counterexample that the nonnegativity assumption $W\geq 0$ cannot be removed.
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