Applications of the division theorem in $H^s$
Résumé
We prove a version of the division theorem in Sobolev spaces
with an estimate of the constant as $s$ tends to infinity. We then
apply it to derive spatial decay estimates for time-periodic
solutions of linear wave equations in one space dimension, and to
prove that the space of decaying solutions is finite-dimensional.
The main point is to show that some of the arguments used to
analyze embedded eigenvalues of Schr\"odinger operators can be
extended to cases where positivity arguments are not available.
This has implications for nonlinear Klein-Gordon equations. A
different approach, based on the proof of the stable manifold
theorem, is also worked out, under slightly different assumptions.