Consider an operator L = \Delta + V on an unbounded cylinder,
where V is a periodic function. As a mapping between unweighted Sobolev
spaces (or Holder spaces), L is not Fredholm. So what are the right
function spaces to work with, where L is Fredholm? When can we solve the
equation Lu = f, with reasonable growth restrictions on u? What are the
general mapping properties of L? To answer these questions, we will use a
gadget called the Fourier-Laplace transform and some contour integrals.