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 $odd\mathrm{.}Thatmeansthat$ V_\varepsilon $isasumofaslowlyvaryingpotential,$ W_0 ( x) $, and one oscillating at frequency $1\mathrm{/}\epsilon$. For $W_0\equiv 0$ we prove that there are no resonances above the line $\text{Im}\lambda =-A\ln ({\epsilon }^{-1})$, except possibly a simple resonance of modulus $\sim {\epsilon }^2$, when $d=1$. We show that this result is optimal by constructing a one-dimensional example. In the case when $W_0\mathrm{\ne }0$ we prove that resonances in fixed strips admit an expansion in powers of $\epsilon$. 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 $\varepsilon \rightarrow 0$andwhoseresonancesapproachresonancesof$V_\varepsilon$modulo$O(\varepsilon^4)$.