H^sup s^-class construction of an almost invariant slow subspace for the Klein–Gordon equation in the non-relativistic limit

We consider the linear Klein–Gordon equation in one spatial dimension with periodic boundary conditions in the non-relativistic limit where ε = ℏ2/(mc2) tends to zero. It is classical that the equation is well posed, for example, in the sense of possessing a continuous semiflow into spaces Hs+1 × Hs for wave function and momentum, respectively. In this paper, we iteratively construct a family of bounded operators FNslow:Hs+1→Hs whose graphs are O(εN)-invariant subspaces under the Klein–Gordon evolution for O(1) times. Contrary to a naive asymptotic series, there is no “loss of derivatives” in the iterative step; i.e., the Sobolev index s can be chosen independent of N. This is achieved by solving an operator Sylvester equation at each step of the construction.