Comparing Sinc and Harmonic Oscillator Basis for Bound States of a Gaussian Interaction

TL;DR

This paper compares sinc collocation and harmonic oscillator bases for solving the Schrödinger equation with a Gaussian potential, analyzing their effectiveness in bound state calculations and proposing corrections for wave function tails.

Contribution

It provides a detailed comparison of the two bases, highlighting their advantages, disadvantages, and proposing a correction for the wave function's asymptotic behavior.

Findings

Sinc basis performs better for both deeply- and weakly-bound states.

Harmonic oscillator basis is more convenient due to orthogonality and mathematical structure.

Large basis size is needed for harmonic oscillator basis to accurately describe both bound states.

Abstract

We investigate the use of the sinc collocation and harmonic oscillator bases for solving a two-particle system bound by a Gaussian potential described by the radial Schr\"odinger equation. We analyze the properties of the bound state wave functions by investigating where the basis-state wave functions break down and relate the breakdowns to the infrared and ultraviolet scales for both bases. We propose a correction for the asymptotic infrared region, the long range tails of the wave functions. We compare the calculated bound state eigenvalues and mean square radii obtained within the two bases. From the trends in the numerical results, we identify the advantages and disadvantages of the two bases. We find that the sinc basis performs better in our implementation for accurately computing both the deeply- and weakly-bound states whereas the harmonic oscillator basis is more convenient since the basis-state wave functions are orthogonal and maintain the same mathematical structure in both position and momentum space. These mathematical properties of the harmonic oscillator basis are especially advantageous in problems where one employs both position and momentum space. The main disadvantage of the harmonic oscillator basis as illustrated in this work is the large basis space size required to obtain accurate results simultaneously for deeply- and weakly-bound states. The main disadvantage of the sinc basis could be the numerical challenges for its implementation in a many-body application.