Scattering from local deformations of a semitransparent plane

Abstract

We study scattering for the couple $(A_F,A_0)$ of Schrödinger operators in $L^2({\mathbb{R}}^3)$ formally defined as $A_0=-\Delta +\alpha {\delta }_{{\pi }_0}$ and $A_F=-\Delta +\alpha {\delta }_{{\pi }_F}$, $\alpha >0$, where ${\delta }_{{\pi }_F}$ is the Dirac $\delta$-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 ${\pi }_0$ 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 $S_F(\lambda )$. Moreover we show that, as $F\to 0$, $\Vert S_F(\lambda )-1{\Vert }_{\mathfrak{B}(L^2({\mathbb{S}}^2))}^2=\mathcal{O}\left({\int }_{{\mathbb{R}}^2}d\text{x}|F(\text{x}){|}^{\gamma }\right)$, $0<\gamma <1$. 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.

We corrected a minor mistake in the computation of the scattering matrix