Instability of Resonances Under Stark Perturbations

Abstract

Let $H^{\varepsilon }=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\varepsilon x +V$, $\varepsilon \ge 0$, 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 $\zeta _n$, ${{\,\mathrm{Im}\,}}\zeta _n<0$, are resonances of $H^{\varepsilon _n}$ for a sequence $\varepsilon _n\downarrow 0$ as $n\rightarrow \infty$ and $\zeta _n\rightarrow \zeta _0$ as $n\rightarrow \infty$, ${{\,\mathrm{Im}\,}}\zeta _0<0$, then $\zeta _0$ is not a resonance of $H^0$.