Scattering from local deformations of a semitransparent plane

Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano

Published 2018-07-20Version 1

We study scattering for the couple $(A_{0},A_{F})$ of Schr\"odinger 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 function $F:\mathbb{R}^{2}\to\mathbb{R}$ and $\pi_{0}$ is the undeformed plane corresponding to the choice $F\equiv 0$. We show asymptotic completeness of the corresponding wave operators, provide a Limiting Absorption Principle and give a representation formula for the Scattering Matrix $S_{F}(\lambda)$. Moreover we show that, as $F\to 0$, $\|S_{F}(\lambda)-\mathsf 1\|^{2}_{\mathfrak{B}(L^{2}({\mathbb S}^{2}))}={\mathcal O}\!\left(\int_{\mathbb{R}^{2}}d\textbf{x}|F(\textbf{x})|^{\gamma}\right)$, $0<\gamma<1$.