Scattering by local deformations of a straight leaky wire

Abstract

We consider a model of a leaky quantum wire with the Hamiltonian $-Δ-αδ(x-Γ)$ in $L^2(\R^2)$, where $Γ$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $Γ$ is a smooth curve and $α\to\infty$.