**Jerome Dorignac
**

We shall review some recent results regarding the properties of quantum breathers in the Quantum DNLS model (also known as Bose-Hubbard model) and its variants. The first part of the talk will be devoted to the localisation properties of quantum breathers. The latter is defined through a two-point density autocorrelation function which is calculated exactly in the two-boson case and evaluated perturbatively (in the weak coupling regime) for N bosons, N>2. We will show that its leading term is exponentially decreasing with the distance between the two points. In the second part of the talk, we will describe the fine structure of some bands of the DNLS spectrum involving states made essentially of two piles of N bosons each (the rest of the lattice being empty). These states are seen as precursors of 2-quantum breathers or two interacting quantum breathers. We will show that the corresponding band is made of [N/2] bound states forming a hierarchy plus a continuum ([] denotes the integral part). Each bound state involves a configuration where the two piles of N bosons are separated by a distance d (in sites). While d < [N/2] bound states exist whereas for d > [N/2], no bound state exists anymore. d=[N/2] seems then a natural interaction length between two quantum breathers containing N bosons each.