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.
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References
Yu.F. Smirnov, K.V. Shitikova, The method of K harmonics and the shell model. Soviet Journal of Particles and Nuclei 8, 344 (1977)
R.D. Amado, H.T. Coelho, \(K\) harmonics in one dimension. Am. J. Phys. 46, 1057 (1978)
W.G. Gibson, S.Y. Larsen, J. Popiel, Hyperspherical harmonics in one dimension: Adiabatic effective potentials for three particles with delta-function interactions. Phys. Rev. A 35, 4919 (1987)
M. Gattobigio, A. Kievsky, M. Viviani, Spectra of helium clusters with up to six atoms using soft-core potentials. Phys. Rev. A 84, 052503 (2011)
N.K. Timofeyuk, Convergence of the hyperspherical-harmonics expansion with increasing number of particles for bosonic systems. Phys. Rev. A 86, 032507 (2012)
N.K. Timofeyuk, Shell model approach to construction of a hyperspherical basis for A identical particles: Application to hydrogen and helium isotopes. Phys. Rev. C 65, 064306 (2002)
N.K. Timofeyuk, Improved procedure to construct a hyperspherical basis for the N-body problem: Application to bosonic systems. Phys. Rev. C 78, 054314 (2008)
D. Baye, P.-H. Heenen, Generalised meshes for quantum mechanical problems. J. Phys. A 19, 2041 (1986)
M. Vincke, L. Malegat, D. Baye, Regularization of singularities in Lagrange-mesh calculations. J. Phys. B 26, 811 (1993)
D. Baye, The Lagrange-mesh method. Phys. Rep. 565, 1 (2015)
D. Baye, M. Hesse, M. Vincke, The unexplained accuracy of the Lagrange-mesh method. Phys. Rev. E 65, 026701 (2002)
P. Descouvemont, C. Daniel, D. Baye, Three-body systems with Lagrange-mesh techniques in hyperspherical coordinates. Phys. Rev. C 67, 044309 (2003)
H. Bateman, Tables of integral transforms, vol. I (McGraw-Hill Book Company, New York, 1954)
M. Abramowitz, I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1965)
D. Baye, Constant-step Lagrange meshes for central potentials. J. Phys. B 28, 4399 (1995)
L. Filippin, M. Godefroid, D. Baye, Relativistic two-photon decay rates with the Lagrange-mesh method. Phys. Rev. A 93, 012517 (2016)
D. Baye, Integrals of Lagrange functions and sum rules. J. Phys. A 44, 395204 (2011)
J.B. McGuire, Study of exactly soluble one-dimensional N-body problem. J. Math. Phys. 5, 622 (1964)
J. Avery, Hyperspherical harmonics and generalized Sturmians (Kluwer Academics Publishers, Dordrecht, 2000)
Yu.F. Smirnov, Talmi transformation for particle with different masses. Nucl. Phys. 39, 346 (1962)
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Timofeyuk, N.K., Baye, D. Hyperspherical Harmonics Expansion on Lagrange Meshes for Bosonic Systems in One Dimension. Few-Body Syst 58, 157 (2017). https://doi.org/10.1007/s00601-017-1318-y
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DOI: https://doi.org/10.1007/s00601-017-1318-y