The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions

Details The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions

2007-09-25

arxiv.org/abs/0709.4022v1

Commun. Math. Phys. 284, 459 (2008)

Physics

Mathematical Physics

LaTeX2e, 19 pages

Scientific paper

10.1007/s00220-008-0521-6

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.

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