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arXiv · 1406.2371

On a problem in eigenvalue perturbation theory

Fritz Gesztesy·Sergey N. Naboko·Roger Nichols
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Abstract

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 $\lambda_0$ 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 $\lambda_0$ 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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Fritz Gesztesy, Sergey N. Naboko, Roger Nichols. 2014-06-09. On a problem in eigenvalue perturbation theory. https://arxiv.org/abs/1406.2371

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