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

R. Seiringer, J. Yin, Communications in Mathematical Physics 284 (2008) 459–479.


Journal Article | Published
Author
Abstract
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-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 ∼ a/r 2 is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼ r -2 above the ground state energy.
Publishing Year
Date Published
2008-12-01
Journal Title
Communications in Mathematical Physics
Volume
284
Issue
2
Page
459 - 479
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Cite this

Seiringer R, Yin J. The Lieb-Liniger model as a limit of dilute bosons in three dimensions. Communications in Mathematical Physics. 2008;284(2):459-479. doi:10.1007/s00220-008-0521-6
Seiringer, R., & Yin, J. (2008). The Lieb-Liniger model as a limit of dilute bosons in three dimensions. Communications in Mathematical Physics, 284(2), 459–479. https://doi.org/10.1007/s00220-008-0521-6
Seiringer, Robert, and Jun Yin. “The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions.” Communications in Mathematical Physics 284, no. 2 (2008): 459–79. https://doi.org/10.1007/s00220-008-0521-6.
R. Seiringer and J. Yin, “The Lieb-Liniger model as a limit of dilute bosons in three dimensions,” Communications in Mathematical Physics, vol. 284, no. 2, pp. 459–479, 2008.
Seiringer R, Yin J. 2008. The Lieb-Liniger model as a limit of dilute bosons in three dimensions. Communications in Mathematical Physics. 284(2), 459–479.
Seiringer, Robert, and Jun Yin. “The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions.” Communications in Mathematical Physics, vol. 284, no. 2, Springer, 2008, pp. 459–79, doi:10.1007/s00220-008-0521-6.

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