Wronskian method and the Schr\"odinger eigenvalue march

TL;DR

This paper compares the Wronskian method and the Schr"odinger eigenvalue march for calculating quantum energies, highlighting how the Wronskian approach offers a rigorous basis and potential efficiency improvements.

Contribution

It introduces a rigorous connection between the Wronskian method and the SEM-CFM, suggesting enhancements for numerical efficiency by incorporating asymptotic wavefunction behavior.

Findings

Wronskian method provides a rigorous foundation for SEM-CFM assumptions.

Asymptotic wavefunction behavior improves numerical efficiency.

Comparison on a simple model demonstrates potential advantages.

Abstract

We compare the Wronskian method (WM) and the Schr\"odinger eigenvalue march or canonical function method (SEM--CFM) for the calculation of the energies and eigenfunctions of the Schr\"odinger equation. The Wronskians between linearly independent solutions of the Schr\"odinger equation provide a rigorous basis for some of the assumptions of the SEM-CFM, like, for example, the concept of "saturation". We compare the performance of both approaches on a simple one-dimensional model and suggest that taking into account the asymptotic behaviour of the wavefunction (as is already done in the WM) may make the SEM-CFM more efficient from a numerical point of view.