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The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
-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 and the radius of the cylindrical trap go to zero, the
Lieb-Liniger model with coupling constant is derived. Our bounds
are uniform in in the whole parameter range , and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size
above the ground state energy.Comment: LaTeX2e, 19 page