Hyperspherical harmonics expansion on Lagrange meshes for bosonic systems in one dimension

TL;DR

This paper develops a hyperspherical harmonics expansion method combined with Lagrange-mesh techniques to study one-dimensional bosonic systems, demonstrating good convergence and accurate energy estimates across various interaction ranges and boson numbers.

Contribution

It introduces an extension of the hyperspherical harmonics expansion with Lagrange-mesh methods for efficient analysis of 1D bosonic systems, including convergence improvements for different interaction ranges.

Findings

Lowest-order energy within 4.5% of exact solutions

Convergence depends on the interaction range, worst for contact interactions

Convergence rate improves with increasing boson number for Gaussian interactions

Abstract

A one-dimensional system of bosons interacting with contact and single-Gaussian forces is studied with an expansion in hyperspherical harmonics. The hyperradial potentials are calculated using the link between the hyperspherical harmonics and the single-particle harmonic-oscillator basis while the coupled hyperradial equations are solved with the Lagrange-mesh method. Extensions of this method are proposed to achieve good convergence with small numbers of mesh points for any truncation of hypermomentum. The convergence with hypermomentum strongly depends on the range of the two-body forces: it is very good for large ranges but deteriorates as the range decreases, being the worst for the contact interaction. In all cases, the lowest-order energy is within 4.5$\%$ of the exact solution and shows the correct cubic asymptotic behaviour at large boson numbers. Details of the convergence studies are presented for 3, 5, 20 and 100 bosons. A special treatment for three bosons was found to be necessary. For single-Gaussian interactions, the convergence rate improves with increasing boson number, similar to what happens in the case of three-dimensional systems of bosons.