A differentiation formula, with application to the two-dimensional Schrödinger equation

TL;DR

The paper tackles accurate discretization of derivatives on grids for problems such as the two-dimensional Schrödinger equation, where standard finite-difference schemes converge slowly due to near-origin behavior. It introduces modified divided differences with respect to an increasing function $\Phi$, deriving first- and second-derivative formulas that are exact for polynomials in the transformed variable $y=\Phi(x)$ and reduce to the ordinary formulas when $\Phi(x)=x$. Applied to the radial Schrödinger equation with $V(r)$, the method demonstrates substantially improved eigenvalues by choosing $\Phi(x)=\sqrt{x}$, and discusses an alternative substitution $\phi(r)=\psi(r)/\sqrt{r}$ as another route to accuracy. The results suggest broad applicability of the approach to quantum-mechanical eigenproblems and other systems with boundary layers or singular behavior near the origin.

Abstract

A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent variable. This method is applied to the numerical solution of the eigenvalue problem for the two-dimensional Schrödinger equation, where standard methods converge very slowly while the approach proposed here gives accurate results.