Quasilinearization Method and Summation of the WKB Series

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 P\"{o}schl-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.