Existence of a resonance (or lack of it) for Schrodinger operators

Question

In dimension 1, consider two potentials $V_1,V_2\in C^\infty(\mathbb{R})$, both of them decaying exponentially fast at $\pm\infty$. Suppose that the Schrodinger operator $$ L_1=-\partial_x^2+V_1, $$ does indeed have a resonance at $\lambda=0$. My question is, if we assume that $V_2$ is sufficiently close to $V_1$, is there any additional hypothesis we can make on $V_2$ to ensure that the operator $$ L_2=-\partial_x^2+V_2, $$ also has a resonance? Here, by "sufficiently close" I mean close in some sufficiently strong norm, let's say $$ \Vert V_1-V_2\Vert_{H^2}\leq \varepsilon $$ for some $0<\varepsilon\ll 1$, for instance. In other words, is there any way to check if a small perturbation of a Schrodinger operator with a resonance also has a resonance?