Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schr"odinger operators with a sufficiently strongly attractive $\delta$ interaction supported by a periodic curve in ${\mathbb{R}}^d,d=2,3$.