计算数学
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计算数学  2018, Vol. 40 Issue (2): 214-226    DOI:
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薛定谔方程的整体几何光学近似
郑春雄
清华大学数学科学系, 北京 100084
GLOBAL GEOMETRICAL OPTICS APPROXIMATION TO THE SCHRODINGER EQUATION
Zhang Chunxiong
Tsinghua University, Department of Mathematical Sciences, Beijing 100084, China
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摘要 整体几何光学方法是一种新的求解高频线性波动方程初值问题的渐进近似理论.该理论最初是对WKB初值数据问题提出来的.在本文中,我们将采用不同的方法,对这一方法予以重新推导,使得该理论同样适用于初值为扩展WKB函数的情形.特别地,我们将建立的理论用于薛定谔方程传播子的半经典近似上来.结果表明,整体几何光学方法提供的波场近似恰好是Kay提出的半相空间公式的一个实例.作为副产品,我们指出Van Vleck近似中起到关键作用的Maslov指标可以通过一个简单的代数关系式来确定.
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关键词薛定谔方程   半经典近似   整体几何光学近似   Maslov指标     
Abstract: Global geometrical optics method is a new asymptotic theory for solving the highfrequency linear wave equations. It was originally proposed for the Cauchy problem with WKB-type initial data. In this paper, we re-deduce this method in a different manner, so that it can be also applied to the Cauchy problem with extended WKB-type initial data. In particular, we apply this theory to the propagator of the Schrödinger equation. It is revealed that the wave-field presented by the global geometrical optics method is exactly the half phase space formula proposed by Kay. As a by-product, we indicate that the Maslov index in the Van Vleck approximation can be determined by a simple algebraic relation through the global geometrical optics approximation.
Key wordsSchrödinger equation   semi-classical approximation   global geometrical optics approximation   Maslov index   
收稿日期: 2017-09-04;
基金资助:

国家自然科学基金(11771248,91630205)资助项目.

引用本文:   
. 薛定谔方程的整体几何光学近似[J]. 计算数学, 2018, 40(2): 214-226.
. GLOBAL GEOMETRICAL OPTICS APPROXIMATION TO THE SCHRODINGER EQUATION[J]. Mathematica Numerica Sinica, 2018, 40(2): 214-226.
 
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