Quasilinearization method and its verification on exactly solvable models in quantum mechanics

Abstract

The proof of the convergence of the quasilinearization method of Bellman and Kalaba, whose origin lies in the theory of linear programming, is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems. This powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on existence of some kind of small parameter. The general properties of the method, particularly its uniform and quadratic convergence, which often also is monotonic, are analyzed and verified on exactly solvable models in quantum mechanics. Namely, application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions. The method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist.

Quasilinearization method and WKB

“…This could explain an extremely sparse use of the technique in physics, where only a few examples of the references to it could be found [5,6,7,8,9]. Recently, however, it was shown [10] by one of the present authors (VBM) that a different proof of the convergence can be provided which allows to extend the applicability of the method to realistic forces defined on infinite intervals with possible singularities at certain points. This proof was generalized and elaborated in the subsequent works [11,12,13,14].…”

“…It was developed originally in theory of linear programming by Bellman and Kalaba [3, 4] to solve nonlinear ordinary and partial differential equations and their systems. In the original works of Bellman and Kalaba [3,4], however, the convergence of the method has been proven only under rather restrictive conditions of small intervals and bounded, nonsingular forces [10] which generally are not fulfilled in physical applications. This could explain an extremely sparse use of the technique in physics, where only a few examples of the references to it could be found [5,6,7,8,9].…”

“…These equations, unlike the nonlinear Calogero equation [5] considered in references [10,12], contain not only quadratic nonlinear terms but various other forms of nonlinearity and not only the first, but also higher derivatives. It was shown that again just a small number of the QLM iterations yield fast convergent and uniformly excellent and stable numerical results.…”

“…If the initial QLM guess is properly chosen the wave function in all QLM iterations, unlike the WKB wave function, is free of unphysical turning point singularities. Since the first QLM iteration is given by an analytic expression [7,8,10,11,12,13], it allows one to analytically estimate the role of different parameters and the influence of their variation on different characteristics of a quantum system. The next iterates display very fast quadratic convergence so that accuracy of energies obtained after a few iterations is extremely high, reaching up to 20 significant figures for a sixth iterate as we show on the example of different widely used physical potentials.…”

“…In the first paper of the series [10], the analytic results of the quasilinearization approach were applied to the nonlinear Calogero equation [5] in the variable phase approach to quantum mechanics, and the results were compared with those of the perturbation theory and with the exact solutions. It was shown that the number of the exactly reproduced perturbation terms doubles with each subsequent QLM iteration, which, of course, is a direct consequence of a quadratic convergence.…”