Scattering resonances for highly oscillatory potentials

Abstract

We study resonances of compactly supported potentials $V_{\epsilon }=W(x,x/\epsilon )$ where $W:{\mathbb{R}}^d\times {\mathbb{R}}^d/(2\pi \mathbb{Z}{)}^d\to \mathbb{C}$, $d$ odd. That means that $V_{\epsilon }$ is a sum of a slowly varying potential, $W_0(x)$, and one oscillating at frequency $1/\epsilon$. 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 $W_0\ne 0$ 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 $\epsilon \to 0$ and whose resonances approach resonances of $V_{\epsilon }$ modulo $O({\epsilon }^4)$.

65 pages; 3 figures. The second version includes numerical computations

References in corpus (1)

• On the spectrum of a Schroedinger operator perturbed by a fast oscillating potential

Cited by in corpus (3)

• Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
• Pollicott-Ruelle resonances via kinetic Brownian motion
• Mathematical Study of Scattering Resonances