Scattering resonances for highly oscillatory potentials

2015/09/14 by Drouot, Alexis
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1509.04198

Abstract

We study resonances of compactly supported potentials V_ε = W ( x, x/ε ) where W : ℝ^d × ℝ^d / ( 2πℤ) ^d → ℂ , d odd. That means that V_ε is a sum of a slowly varying potential, W₀ ( x) , and one oscillating at frequency 1/ε. For W₀ ≡ 0 we prove that there are no resonances above the line Im λ= -A ln(ε^-1), except possibly a simple resonance of modulus ∼ ε², when d=1. We show that this result is optimal by constructing a one-dimensional example. In the case when W₀ ≠ 0 we prove that resonances in fixed strips admit an expansion in powers of ε. The argument provides a method for computing the coefficients of the expansion. In particular we produce an effective potential converging uniformly to W₀ as ε → 0 and whose resonances approach resonances of V_ε modulo O(ε⁴).

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