Few-Body Systems

, 58:157 | Cite as

Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension

  • N. K. Timofeyuk
  • D. Baye
Open Access


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.


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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of SurreyGuildfordEngland, UK
  2. 2.Physique Quantique and Physique Nucléaire Théorique et Physique Mathématique, C.P. 229Université Libre de Bruxelles (ULB)BrusselsBelgium

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