Quasilinearization method and WKB

Abstract

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers of , the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schrödinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.

Initial time difference quasilinearization for Caputo Fractional Differential Equations

“…Also this method has been generalized, refined and extended in several directions so as to be applicable to a much larger class of nonlinear problems by not demanding convexity or concavity property. Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications [2][3][4][5][6][7][8].…”

Unified QLM for regular and arbitrary singular potentials

“…For regular potentials, the numerical implementation has been completed in the work [18], where the Langer WKB solution [20] was found to assure immediate onset of quadratic convergence of the QLM iteration.…”

“…It is applied in quantum mechanics by rewriting the radial Schrödinger equation as a Riccati equation for a function expressed in terms of the logarithmic derivative /ðrÞ of the wave function. QLM is a resummation of WKB: the kth QLM iteration sums 2 k terms [17,18] of the WKB series. For exactly solvable potentials the first QLM iteration gives exact energies if a quantization condition is imposed [19].…”

“…Our numerical implementation for the radial Schrödinger equation [14,17,18,21], for easier handling of nodes in the radial solution vðrÞ, uses the (negative) form of solution:…”

Approximation of Solutions for Second-Order m-Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization

“…To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization 10 . This method has been developed for a variety of problems [11][12][13][14][15][16][17][18][19][20] . In view of its diverse applications, this approach is quite an elegant and easier for application algorithms.…”