Instability of resonances under Stark perturbations

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 $H^0$.