Quasilinearization Method and Summation of the WKB Series

TL;DR

This paper compares the quasilinearization method with WKB solutions, showing that QLM reproduces WKB series terms and yields highly accurate energy levels for various potentials, converging rapidly.

Contribution

It demonstrates that QLM reproduces WKB series terms and provides highly accurate solutions for quantum potentials, with rapid convergence and exact energies for certain cases.

Findings

QLM reproduces the structure of the WKB series.

Exact energies obtained for several potentials.

Rapid convergence of QLM iterates, achieving high precision.

Abstract

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.