Instability of resonances under Stark perturbations

TL;DR

This paper investigates how resonances in quantum systems with a linear Stark potential and a finite-rank perturbation behave as the Stark field vanishes, showing that resonances do not persist in the limit.

Contribution

It demonstrates that resonances of the perturbed Hamiltonian do not converge to resonances of the unperturbed Hamiltonian as the Stark field approaches zero.

Findings

Resonances do not persist in the zero Stark field limit.

Resonances of $H^{ ext{epsilon}}$ do not converge to those of $H^0$.

Resonance positions are unstable under Stark perturbations.

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 $H^0$.