Eigenvalues from power--series expansions: an alternative approach

TL;DR

This paper introduces an alternative rational approximation method for calculating eigenvalues of anharmonic oscillators, demonstrating faster convergence than traditional power-series approaches.

Contribution

It presents a novel rational approximation technique that improves the accuracy and convergence rate in eigenvalue calculations for Schrödinger equations.

Findings

Faster convergence compared to Hill-determinant method

Accurate eigenvalues obtained via roots of Hankel determinants

Effective for anharmonic oscillator eigenfunctions

Abstract

An appropriate rational approximation to the eigenfunction of the Schr\"{o}dinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of Hankel determinants. The convergence rate of this approach is greater than that for a well--established method based on a power--series expansions weighted by a Gaussian factor with an adjustable parameter (the so--called Hill--determinant method).