Normal form and long time analysis of splitting schemes for the linear Schr?dinger equation with small potential

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• 作者：Guillaume Dujardin and Erwan Faou
• 关键词：Mathematics Subject Classification (2000) 65P10 ; 37M15 ; 37K55
• 刊名：Numerische Mathematik
• 出版年：2007
• 出版时间：December, 2007
• 年：2007
• 卷：108
• 期：2
• 页码：223-262
• 全文大小：789.3 KB

文摘

We consider the linear Schr?dinger equation on a one dimensional torus and its time-discretization by splitting methods. Assuming a non-resonance condition on the stepsize and a small size of the potential, we show that the numerical dynamics can be reduced over exponentially long time to a collection of two dimensional symplectic systems for asymptotically large modes. For the numerical solution, this implies the long time conservation of the energies associated with the double eigenvalues of the free Schr?dinger operator. The method is close to standard techniques used in finite dimensional perturbation theory, but extended here to infinite dimensional operators.