H 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...
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Veröffentlicht in: | Journal of mathematical physics 2018-05, Vol.59 (5) |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Online-Zugang: | Volltext |
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Zusammenfassung: | 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 FslowN: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. |
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ISSN: | 0022-2488 1089-7658 |
DOI: | 10.1063/1.5027040 |