Transformation of perturbation series into continued fractions, with application to an anharmonic oscillator

An analytic iterative approach to solving the time‐independent Schrödinger equation

“…The perturbative energy levels were first found using Padé approximants by Reid [6]. Numerous other methods, both analytic and numerical, have been used in an effort to establish methods for solving this problem.…”

“…We follow the works of others [5][6][7] and set the following: ϭ 2, m ϭ 0.5, and ϭ {0.1, 0.2, 0.3, 1, 2, 3}.…”

Asymptotic response of observables from divergent weak-coupling expansions: A fractional-calculus-assisted Padé technique

“…Usually, form (1) is obtained from the Rayleigh-Schrödinger perturbation theory and (2) is intuitively obvious. In a few situations, however, form (2) stands for the leading behavior; in actuality, it is replaced by a strong convergent expansion at very large x. In cases of external perturbations, the coupling parameter x can be varied.…”

“…Ready examples include the variations of energy of an atom under high electric and/or magnetic fields [1]. In dealing with some problems, there exist alternative routes of arriving at (2) [e.g., variational], but not always. For instance, the variation method does not apply if the corresponding system Hamiltonian is not bounded from below.…”

On the numerical solution of the Schr�dinger equation with a polynomial potential

“…Among these, the works of Hioe and Montroll [3] proposed a truly efficient algorithm for obtaining the eigenvalues and gave extensive tables for different values of the perturbation parameter. Henkel and Uzes [4] approached the problem from the point of view of creation and annihilation op-erators, and Reid [5] used an approximation method based on rational fractions. Hill [6] developed an ingenious method to calculate upper and lower bounds of the eigenvalues, getting an accuracy of 21 figures when locating the eigenvalue between these bounds.…”