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
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

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.