Soft isogeometric analysis of the Bound States of a Quantum Three-Body Problem in 1D

TL;DR

This paper introduces softIGA, a novel variation of isogeometric analysis, which reduces spectral errors in quantum three-body bound state calculations in 1D, outperforming traditional IGA and BO approximation.

Contribution

The paper develops and applies softIGA with high-order derivative-jump penalties to improve accuracy in quantum three-body problems, demonstrating eigenvalue superconvergence.

Findings

SoftIGA reduces spectral errors in low-energy quantum states.

Optimal penalty parameters lead to eigenvalue superconvergence.

SoftIGA outperforms classical IGA and BO approximation in examples.

Abstract

The study of quantum three-body problems has been centered on low-energy states that rely on accurate numerical approximation. Recently, isogeometric analysis (IGA) has been adopted to solve the problem as an alternative but more robust (with respect to atom mass ratios) method that outperforms the classical Born-Oppenheimer (BO) approximation. In this paper, we focus on the performance of IGA and apply the recently-developed softIGA to reduce the spectral errors of the low-energy bound states. The main idea is to add high-order derivative-jump terms with a penalty parameter to the IGA bilinear forms. With an optimal choice of the penalty parameter, we observe eigenvalue error superconvergence. We focus on linear (finite elements) and quadratic elements and demonstrate the outperformance of softIGA over IGA through a variety of examples including both two- and three-body problems in 1D.