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H. Mohamad and M. Oliver,
\(H^s\)-class construction of an almost invariant slow subspace
for the Klein-Gordon equation in the non-relativistic limit,
J. Math. Phys. 59 (2018), 051509.
Abstract:
We consider the linear Klein--Gordon equation in one spatial dimension
with periodic boundary conditions in the non-relativistic limit where
\(\varepsilon = \hbar^2/(mc^2)\) tends to zero. It is classical that the
equation is well posed, for example, in the sense of possessing a
continuous semiflow into spaces \(H^{s+1} \times H^{s}\) for wave
function and momentum, respectivly. In this paper, we iteratively
contruct a family of bounded operators
\(F^N \colon H^{s+1} \to H^{s}\) whose graphs are
\(O(\varepsilon^N)\)-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.
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