Soft IsoGeometric analysis of the bound states of a quantum three-body problem in 1D

Abstract

The study of quantum three-body problems has been centred 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, especially for the cases with small mass ratios. In this paper, we focus on the performance of IGA and apply the recently-developed softIGA to further reduce the spectral errors of the low-energy bound states. This is an extension to the recent work that was published as an ICCS conference paper in Deng (2022). The main idea of softIGA 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. Herein, the optimal parameter coincides with the ones for the Laplace operator (zero potential) and can be heuristically computed for a general elliptic operator. We focus on linear 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.