Scattering on leaky wires in dimension three

We consider the scattering problem for a class of strongly singular Schr"odinger operators in $L^2(\mathbb{R}{R}^3)$ which can be formally written as $H_{\alpha ,\mathrm{\Gamma }}=-\mathrm{\Delta }+{\delta }_{\alpha }(x-\mathrm{\Gamma })$ where $\alpha \in \mathbb{R}$ is the coupling parameter and $\mathrm{\Gamma }$ is an infinite curve which is a local smooth deformation of a straight line $\mathrm{\Sigma }\subset {\mathbb{R}}^3$. Using Kato-Birman method we prove that the wave operators ${\mathrm{\Omega }}_{\pm }(H_{\alpha ,\mathrm{\Gamma }},H_{\alpha ,\mathrm{\Sigma }})$ exist and are complete.