Instability of resonances under Stark perturbations

Abstract

Let $H^{\epsilon }=-\frac{d^2}{d{x}^2}+\epsilon x+V$, $\epsilon \ge 0$, on $L^2(\mathbf{R})$. Let $V={\sum }_{k=1}^N{c}_k|{\psi }_k⟩⟨{\psi }_k|$ 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 $H^{{\epsilon }_n}$ for a sequence ${\epsilon }_n\downarrow 0$ as $n\to \infty$ and ${\zeta }_n\to {\zeta }_0$ as $n\to \infty$, $\mathrm{Im}{\zeta }_0<0$, then ${\zeta }_0$ is \emph{not} a resonance of $H^0$.

11 pages

References in corpus (2)

• Asymptotics of resonances for 1d Stark operators
• Instability of pre-existing resonances under a small constant electric field