Scattering resonances for highly oscillatory potentials

We study resonances of compactly supported potentials $V_\varepsilon = W ( x, x/\varepsilon )$ where $W : \mathbb{R}^d \times \mathbb{R}^d / ( 2\pi \mathbb{Z}) ^d \to \mathbb{C}$, $d$ odd. That means that $V_\varepsilon$ is a sum of a slowly varying potential, $W_0 ( x)$, and one oscillating at frequency $1/\varepsilon$. For $W_0 \equiv 0$ we prove that there are no resonances above the line $\text{Im} \lambda = -A \ln(\varepsilon^{-1})$, except possibly a simple resonance of modulus $\sim \varepsilon^2$, when $d=1$. We show that this result is optimal by constructing a one-dimensional example. In the case when $W_0 \neq 0$ we prove that resonances in fixed strips admit an expansion in powers of $\varepsilon$. The argument provides a method for computing the coefficients of the expansion. In particular we produce an effective potential converging uniformly to $W_0$ as $\varepsilon \rightarrow 0$ and whose resonances approach resonances of $V_\varepsilon$ modulo $O(\varepsilon^4)$.