Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators

TL;DR

This paper demonstrates a conformal mapping method for solving nonlinear ODEs, specifically anharmonic oscillators, showing high accuracy and efficiency compared to other power series methods, especially for certain potential parameters.

Contribution

It introduces and illustrates a conformal mapping technique for solving boundary value problems in nonlinear ODEs, improving convergence and accuracy over traditional power series approaches.

Findings

High accuracy results for positive and slightly negative potential parameters

Method's efficiency decreases as the potential parameter becomes more negative

Comparison shows advantages over similar Taylor series based methods

Abstract

The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The method [Nucl. Phys. B801, 296 (2008)], applies a priori to any ODE with two-point boundaries (one being located at infinity), the solution of which has singularities in the complex plane of the independent variable $x$. A conformal mapping of a suitably chosen angular sector of the complex plane of $x$ upon the unit disc centered at the origin makes convergent the transformed Taylor series of the generic solution so that the boundary condition at infinity can be easily imposed. In principle, this constraint, when applied on the logarithmic-derivative of the wave function, determines the eigenvalues to an arbitrary level of accuracy. In practice, for $\beta \geq 0$ or slightly negative, the accuracy of the results obtained is astonishingly large with regards to the modest computing power used. It is explained why the efficiency of the method decreases as $\beta $ is more and more negative. Various aspects of the method and comparisons with some seemingly similar methods, based also on expressing the solution as a Taylor series, are shortly reviewed, presented and discussed.