拟线性化方法与WKB级数的求和
Quasilinearization Method and Summation of the WKB Series
摘要：
通过准线性化方法（QLM）得到的解决方案进行了比较与WKB解决方案。以$\hbar$的幂对第$p$个QLM项进行扩展再现了WKB级数的结构，生成了无限数量的WKB项，其中第一个$2^p$项被精确再现。QLM量子化条件导致确切的能量Pöschl-泰勒，Hulthen，Hylleraas，莫尔斯，Eckart潜力等其他更复杂的潜力的第一QLM，给出了封闭的解析表达式，是非常准确的。迭代收敛得非常快。非简谐振子和二体库仑狄拉克方程的能量的第六阶精度为20位有效数字。
英文摘要：
Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the $p$-th QLM iterate in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first $2^p$ terms reproduced exactly. The QLM quantization condition leads to exact energies for the Pöschl-Teller, Hulthen, Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb Dirac equation has an accuracy of 20 significant figures.
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