Stark扰动下共振态的不稳定性
Instability of resonances under Stark perturbations
摘要：
设$H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$、$\varepsilon\geq0$、在$L^2(\mathbf{R})$上。令$V=\sum_{k=1}^Nc_k|ψ_k\rangle\langleψ_k|$是秩$N$运算符，其中$ψ_k\in L^2(\mathbf{R})$是实数、支持complex且为偶数。共振定义使用解析散射理论。主要结果是，如果$ζ_n$，${\rm Im}ζ_n<0$，是$H^{\varepsilon_n}$的共振，对于序列$\varepsilon_n\downarrow0$作为$n\to\infty$，$ζ_n\toζ_0$作为$n\to\infty$，${\rm Im}ζ_0<0$，则$ζ_0$是\emph{not}，$H^0$的共振。
英文摘要：
Let $H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$, $\varepsilon\geq0$, on $L^2(\mathbf{R})$. Let $V=\sum_{k=1}^Nc_k|ψ_k\rangle\langleψ_k|$ be a rank $N$ operator, where the $ψ_k\in L^2(\mathbf{R})$ are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if $ζ_n$, ${\rm Im}ζ_n<0$, are resonances of $H^{\varepsilon_n}$ for a sequence $\varepsilon_n\downarrow0$ as $n\to\infty$ and $ζ_n\toζ_0$ as $n\to\infty$, ${\rm Im}ζ_0<0$, then $ζ_0$ is \emph{not} a resonance of $H^0$.
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