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.
Highlights
The expansion in hyperspherical harmonics (HH) of a many-body wave function is a powerful scheme that allows the Schrödinger equation to be solved in a systematic way by reducing it to a system of coupled equations of only one dynamic variable for any number of particles N [1]
This seems to contradict more recent observations made for three-dimensional systems that the HH expansion for a single-Gaussian two-body potential converges better with larger N [4,5]
The hyperspherical harmonics are expressed via linear combinations of symmetrized products of single-particle states that contain only the lowest-energy centre-of-mass motion and do not contain hyperradial excitations
Summary
The expansion in hyperspherical harmonics (HH) of a many-body wave function is a powerful scheme that allows the Schrödinger equation to be solved in a systematic way by reducing it to a system of coupled equations of only one dynamic variable for any number of particles N [1]. The authors point out that the N -dependence of the lowest-order energy is quadratic while the exact solution gives a cubic N -dependence This seems to contradict more recent observations made for three-dimensional systems that the HH expansion for a single-Gaussian two-body potential converges better with larger N [4,5]. This means that the lowest-order energy should get closer to the exact solution with increasing N , which was checked numerically in Ref.
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