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

  • Robert Seiringer
  • Jun Yin
Article

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.

Keywords

Ground State Energy Scattering Length Cylindrical Trap Impenetrable Boson Effective Coupling Parameter 
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Copyright information

© The Authors 2008

Authors and Affiliations

  1. 1.Department of Physics, Jadwin HallPrinceton UniversityPrincetonUSA

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