Lieb-Liniger delta Bose gas

introduction

• N bosons interacting on a line of length L via the delta function potential
  
• one-dimensional Bose gas
  
• 1963 Lieb and Liniger solved by Bethe ansatz
  

 

Hamiltonian

• quantum mechanical Hamiltonian

\[H=-\sum_{j=1}^{N}\frac{\partial^2}{\partial x_j^2}+2c\sum_{1\leq i<j\leq N}^{N}\delta(x_i-x_j)\]
 

two-body scattering term

• \(s_{ab}=k_a-k_b+ic\)
  

Bethe-ansatz equation

\[\exp(ik_jL)=\prod_{l=1}^{N}\frac{k_j-k_l+ic}{k_j-k_l-ic}\]

 

energy spectrum

• energy of a Bethe state

\[E=\sum_{j=1}^{N}k_j^2\]

 

encyclopedia

• http://en.wikipedia.org/wiki/Lieb-Liniger_model

 

articles

• C. N. Yang and C. P. Yang Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction, J. Math. Phys. 10, 1115 (1969)
• C.N. Yang Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315
• Elliott H. Lieb and Werner Liniger Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, 1963