#### Vol. 4, No. 3-4, 2016

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A remark on eigenvalue perturbation theory at vanishing isolation distance

### Fiorella Barone and Sandro Graffi

Vol. 4 (2016), No. 3-4, 297–309
##### Abstract

Let $T$ be a self-adjoint operator in a separable Hilbert space $X$, admitting compact resolvent and simple eigenvalues with possibly vanishing isolation distance, and let $V$ be symmetric and bounded. Consider the self-adjoint operator family $T\left(g\right):g\in ℝ$ in $X$ defined by $T+gV$ on $D\left(T\right)$. A simple criterion is formulated ensuring, for any eigenvalue of $T\left(g\right)$, the existence to all orders of its perturbation expansion and its asymptotic nature near $g=0$, with estimates independent of the eigenvalue index. An application to a class of Schrödinger operators is described.

##### Keywords
isolation distance, eigenvalue perturbation theory
##### Mathematical Subject Classification 2010
Primary: 81Q05, 81Q10, 81Q15