Quasilinear Approach to Summation of the WKB Series

Abstract

It is shown that the quasilinearization method (QLM) sums the WKB series. The method approaches solution of the Riccati equation (obtained by casting the Schr\"{o}dinger 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, P\"{o}schl-Teller, Hulthen, Hyleraas, Morse, Eckart etc.