  We consider a model of leaky quantum wire in three dimensions. The
Hamiltonian is a singular perturbation of the Laplacian supported by a line
with the coupling which is bounded and periodically modulated along the line.
We demonstrate that such a system has a purely absolutely continuous spectrum
and its negative part has band structure with an at most finite number of gaps.
This result is extended also to the situation when there is an infinite number
of the lines supporting the perturbations arranged periodically in one
direction.
