Solving Schr\"odinger's equation by B-spline collocation

TL;DR

This paper investigates the use of Gaussian collocation points in B-spline methods for solving Schrödinger's equation, demonstrating improved accuracy in many cases, especially for hydrogen atom wave functions.

Contribution

It introduces the application of Gaussian points in B-spline collocation for Schrödinger's equation, showing their effectiveness over traditional point distributions in certain scenarios.

Findings

Gaussian collocation points often outperform equally spaced points.

Performance varies depending on mesh size and problem specifics.

Gaussian points are highly effective for hydrogen atom wave functions.

Abstract

B-spline collocation techniques have been applied to Schr\"odinger's equation since the early 1970s, but one aspect that is noticeably missing from this literature is the use of Gaussian points (i.e., the zeros of Legendre polynomials) as the collocation points, which can significantly reduce approximation errors. Authors in the past have used equally spaced or nonlinearly distributed collocation points (noticing that the latter can increase approximation accuracy) but, strangely, have continued to avoid Gaussian collocation points. Using the methodology and computer routines provided by Carl de Boor's book A Practical Guide to Splines as a `numerical laboratory', the present dissertation examines how the use of Gaussian points can interact with other features such as box size, mesh size and the order of polynomial approximants to affect the accuracy of approximations to Schr\"odinger's bound state wave functions for the electron in the hydrogen atom. We explore whether or not, and under what circumstances, B-spline collocation at Gaussian points can produce more accurate approximations to Schr\"odinger's wave functions than equally spaced and nonlinearly distributed collocation points. We also apply B-spline collocation at Gaussian points to a Schr\"odinger equation with cubic nonlinearity which has been used extensively in the past to study nonlinear phenomena. Our computer experiments show that collocation at Gaussian points can be a highly successful approach for the hydrogen atom, consistently superior to equally spaced collocation points and often superior to nonlinearly distributed collocation points. However, we do encounter some situations, typically when the mesh is quite coarse relative to the box size for the hydrogen atom, and also in the cubic Schr\"odinger equation case, in which nonlinearly distributed collocation points perform better than Gaussian collocation points.