Numerological analysis of the WKB approximation in large order
Numerological analysis of the WKB approximation in large order
复制标题
大阶 WKB 近似的数值分析
DOI：
10.1103/physrevd.16.1740
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发表时间：
1977
影响因子：
5
通讯作者：
Paul S. Wang
中科院分区：
文献类型：
--
作者：
C. Bender;K. Olaussen;Paul S. Wang
我们展示了如何在WKB近似下求解解析势的一维双转折点本征值问题到所有阶数。我们用这种方法计算了$x^N$（$N$为偶数）势到12阶的本征值。对于$x^4$势，第10个本征值的数值结果精确到$10^{15}$分之一。对于$\nu_0\cosh^{-2}x$势，WKB级数简化为一个几何级数，可以求和得到精确答案。最后，我们报告了关于WKB级数结构的数字学实验结果。我们结果的简洁性使我们（不太确定地）推测，对于任意解析势，有可能找到WKB级数各项的公式。
We show how to slove the one-dimensional two-turning-point eigenvalue problem for analytic potentials to all orders in the WKB approximation. We use this method to compute the eigenvalues of the ${x}^{N}$ ($N$ even) potential to twelfth order. Numerical results for the ${x}^{4}$ potential are accurate to 1 part in ${10}^{15}$ for the tenth eigenvalue. For the ${\ensuremath{\nu}}_{0}{cosh}^{\ensuremath{-}2}x$ potential the WKB series reduces to a geometric series which may be summed to give the exact answer. Finally, we report on the results of numerological experiments on the structure of the WKB series. The simplicity of our results leads us to conjecture (weakly) that it may be possible to find a formula for the terms of the WKB series for arbitrary analytic potentials.