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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schroedinger equation
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Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schroedinger equation

机译：带有极点的鸭的渐近展开。在平稳一维薛定inger方程中的应用

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摘要

The central topic of this paper is the problem of turning points. The paradigm is the stationary unidimensional Schroedinger equation, with various potentials. The first step is to transform the linear equation of second order into a Riccati equation. The non standard analysis and the theory of canards allow to compute the first eigenvalue and the corresponding solution. With a change of variables, it is possible to reduce the problem of the n-th energy level to the (n—1)-th. The first result (already proved by others methods) of the paper is an algorithm to compute the asymptotic expansion of the n-th energy level in powers of the Planck's constant. The second (new) result is an algorithm to compute an expansion of the corresponding solution. This expansion is a fraction so that the singularity is resolved. For example it is possible to determine the zero of the eigenfunctions of the Schroedinger operator up to any power of the Planck's constant. The algorithms are implemented in Maple, and illustrated with a double symmetrical well as potential.
机译：本文的中心主题是转折点问题。范例是具有各种电势的平稳一维Schroedinger方程。第一步是将二阶线性方程式转换为Riccati方程式。非标准分析和卡纳德理论允许计算第一个特征值和相应的解。通过改变变量，可以将第n个能级的问题减少到第（n-1）个。该论文的第一个结果（已经由其他方法证明）是一种以普朗克常数的幂计算第n个能级的渐近展开的算法。第二个（新）结果是一种算法，用于计算相应解的扩展。该扩展是分数，因此可以解决奇点问题。例如，有可能确定Schroedinger算子的本征函数的零值，直至普朗克常数的任意幂。该算法在Maple中实现，并以双对称井和势进行了说明。

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