Scattering from local deformations of a semitransparent plane

We study scattering for the couple (AF,A0) of Schrödinger operators in L2(R3) formally defined as A0=−Δ+αδπjavax.xml.bind.JAXBElement@51351cec and AF=−Δ+αδπjavax.xml.bind.JAXBElement@618cd9ea, α>0, where δπjavax.xml.bind.JAXBElement@2888f896 is the Dirac δ-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function F:R2→R and π0 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 SF(λ). Moreover we show that, as F→0, ‖SF(λ)−1‖B(Ljavax.xml.bind.JAXBElement@4800b62e(Sjavax.xml.bind.JAXBElement@312b11be))2=O(∫Rjavax.xml.bind.JAXBElement@366fcdf8dx|F(x)|γ), 0<1.