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arXiv · 1804.05620

Instability of resonances under Stark perturbations

Arne Jensen·Kenji Yajima
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Abstract

Let $H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$, $\varepsilon\geq0$, on $L^2(\mathbf{R})$. Let $V=\sum_{k=1}^Nc_k|\psi_k\rangle\langle\psi_k|$ be a rank $N$ operator, where the $\psi_k\in L^2(\mathbf{R})$ are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if $\zeta_n$, ${\rm Im}\zeta_n<0$, are resonances of $H^{\varepsilon_n}$ for a sequence $\varepsilon_n\downarrow0$ as $n\to\infty$ and $\zeta_n\to\zeta_0$ as $n\to\infty$, ${\rm Im}\zeta_0<0$, then $\zeta_0$ is \emph{not} a resonance of $H^0$.

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Arne Jensen, Kenji Yajima. 2018-04-16. Instability of resonances under Stark perturbations. https://doi.org/10.1007/s00023-018-0746-7

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