Scattering from local deformations of a semitransparent plane

We study scattering for the couple (A F ,A 0 ) of Schrödinger operators in L 2 (R 3 ) formally defined as A 0 =−Δ+αδ π javax.xml.bind.JAXBElement@35266ea2 and A F =−Δ+αδ π javax.xml.bind.JAXBElement@56dd5bc1 , α>0, where δ π javax.xml.bind.JAXBElement@4dbab1e is the Dirac δ-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function F:R 2 →R and π 0 is the undeformed plane corresponding to the choice F≡0. We provide a Limiting Absorption Principle, show asymptotic completeness of the wave operators and give a representation formula for the corresponding Scattering Matrix S F (λ). Moreover we show that, as F→0, ‖S F (λ)−1‖ B(L javax.xml.bind.JAXBElement@b81c41d (S javax.xml.bind.JAXBElement@dd4f8b1 )) 2 =O(∫ R javax.xml.bind.JAXBElement@316bb999 dx|F(x)| γ ), 0<γ<1.