The one-dimensional delta Bose gas, also known as the Lieb-Liniger model, is instrumental in the description of one-dimensional ultra-cold quantum gases. In its homogeneous, translation-invariant version, a key property of the model is that it is solvable by Bethe Ansatz. However, in a trap potential, this property is lost. I will give an introduction to the results that have accumulated recently on correlation functions and on the dynamics of the inhomogeneous 1d delta Bose gas. The results are obtained from large-scale approaches that are valid in those situations where the external potential and the local thermodynamic quantities vary slowly at the inter particle distance

Université de Lorraine and CNRS