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 $H_{α,Γ}= -Δ+ δ_α(x-Γ)$ where $α\in\mathbb{R}$ is the coupling parameter and $Γ$ is an infinite curve which is a local smooth deformation of a straight line $Σ\subset\mathbb{R}^3$. Using Kato-Birman method we prove that the wave operators $Ω_\pm(H_{α,Γ}, H_{α,Σ})$ 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