WKB级数求和的拟线性方法
Quasilinear Approach to Summation of the WKB Series
摘要：
证明了拟线性化方法（QLM）对WKB级数求和。该方法通过用一系列线性项逼近非线性项来逼近Riccati方程（将薛定谔方程转换为非线性形式）的解，而不是基于小参数的存在。每一个p阶QLM积分都可以用一个封闭的积分形式来表示。它在$\hbar$的幂上的展开再现了WKB级数的结构，产生了无限数量的WKB项。展开式的第一个$2^p$项的系数是精确的，而类似数目的下一项的系数是近似的。在任何QLM迭代中的量子化条件，包括第一次，导致许多众所周知的物理势，如库仑，谐振子，Pöschl-Teller，Hulthen，Hyleraas，Morse，Eckart等的精确能量。
英文摘要：
It is shown that the quasilinearization method (QLM) sums the WKB series. The method approaches solution of the Riccati equation (obtained by casting the Schrödinger equation in a nonlinear form) by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of a smallness parameter. Each p-th QLM iterate is expressible in a closed integral form. Its expansion in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms. Coefficients of the first $2^p$ terms of the expansion are exact while coefficients of a similar number of the next terms are approximate. The quantization condition in any QLM iteration, including the first, leads to exact energies for many well known physical potentials such as the Coulomb, harmonic oscillator, Pöschl-Teller, Hulthen, Hyleraas, Morse, Eckart etc.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
