Scattering from local deformations of a semitransparent plane

Abstract

We study scattering for the couple (AF , A0) of Schrödinger operators in L 2 (R 3 ) formally defined as A0 = −∆ + α δπ 0 and AF = −∆ + α δπ F , α > 0, where δπ F 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 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 2 B(L 2 (S 2 )) = O R 2 dx|F (x)| γ , 0 < γ < 1.