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.