解常微分方程的保角映射与其它幂级数方法的比较：以非谐振子为例
Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators
摘要：
通过对具有势$V(x) =βx^{2}+x^{2m}$（$m>0$）的非谐振子（AO）的研究，说明了求解非线性常微分方程（ODE）的准解析方法的简单性和有效性.方法[Nucl.Phys.B801，296（2008）]先验地应用于具有两点边界（一个位于无穷远）的任何常微分方程，其解在自变量$x$的复平面中具有奇点。将$x$复平面的一个适当选择的角扇区保角映射到以原点为中心的单位圆盘上，使得一般解的变换泰勒级数收敛，从而可以容易地施加无穷远处的边界条件。原则上，当应用于波函数的微分时，该约束将本征值确定为任意精度水平。在实践中，对于$β\geq 0$或略负的情况，所获得的结果的准确性相对于所使用的适度计算能力而言是非常大的。解释了当$β$越来越负时，该方法的效率降低的原因。该方法的各个方面和比较与一些看似类似的方法，也表示为泰勒级数的解决方案的基础上，简短地审查，介绍和讨论。
英文摘要：
The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =βx^{2}+x^{2m}$ ($m>0$). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable $x$. A conformal mapping of a suitably chosen angular sector of the complex plane of $x$ upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for $β\geq 0$ or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as $β$ is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.
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