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Scattering from local deformations of a semitransparent plane

We study scattering for the couple $(A_{F},A_{0})$ of Schrödinger operators in $L^2(\mathbb{R}^3)$ formally defined as $A_0 = -Δ+ α\, δ_{π_0}$ and $A_F = -Δ+ α\, δ_{π_F}$, $α>0$, where $δ_{π_F}$ is the Dirac $δ$-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function $F:\mathbb{R}^{2}\to\mathbb{R}$ and $π_{0}$ is the undeformed plane corresponding to the choice $F\equiv 0$. 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 $F\to 0$, $\|S_{F}(λ)-\mathsf 1\|^{2}_{\mathfrak{B}(L^{2}({\mathbb S}^{2}))}={\mathcal O}\!\left(\int_{\mathbb{R}^{2}}d\textbf{x}|F(\textbf{x})|^γ\right)$, $0<γ<1$. We correct a minor mistake in the computation of the scattering matrix, occurring in the published version of this paper (see J. Math. Anal. Appl. 473(1) (2019), pp. 215-257). The mistake was in Section 7, and affected the statement of Corollary 7.2, specifically, Eq. (7.8). Regrettably the formula for $S_F$ in the Corrigendum J. Math. Anal. Appl. 482(1) (2020), 123554, still contains a misprint, the correct expression is the one given here.