Scattering on leaky wires in dimension three

Abstract

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

11 pages, to appear in "Analysis and Operator Theory. Dedicated in Memory of Tosio Kato 100th Birthday" (Themistocles M. Rassias and Valentin A. Zagrebnov, eds.), Springer