The Lieb-Liniger model as a limit of dilute bosons in three dimensions

Robert Seiringer Department of Physics, Jadwin Hall, Princeton University Jun Yin Department of Physics, Jadwin Hall, Princeton University

Mathematical Physics mathscidoc:1608.22015

Communications in Mathematical Physics , 284, (2), 459-479, 2008
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive $\delta$-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length $a$ and the radius $r$ of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant $g \sim a/r^2$ is derived. Our bounds are uniform in $g$ in the whole parameter range $0\leq g\leq \infty$, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size $\sim r^{-2}$ above the ground state energy.
Lieb-Liniger model, dilute bosons
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@inproceedings{robert2008the,
  title={The Lieb-Liniger model as a limit of dilute bosons in three dimensions},
  author={Robert Seiringer, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824110101616786432},
  booktitle={Communications in Mathematical Physics },
  volume={284},
  number={2},
  pages={459-479},
  year={2008},
}
Robert Seiringer, and Jun Yin. The Lieb-Liniger model as a limit of dilute bosons in three dimensions. 2008. Vol. 284. In Communications in Mathematical Physics . pp.459-479. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824110101616786432.
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