Quantum bound states and matter waves delocalizations

We discuss localized ground states of the periodic Gross-Pitaevskii equation (GPE) in the framework of a quantum linear Schrodinger equation with effective potential determined in self-consistent manner. We show that depending on the interaction among the atoms being attractive or repulsive, bound states of the linear self-consistent problem are formed in the forbidden zones of the linear spectrum below or above the energy bands. These bound states correspond to exact solitons of the GPE equation. The implication of these results on delocalizing transitions of multidimensional solitons of the GPE is also discussed. In particular we show that the delocalizing transition corresponds to the threshold for the existence of a single bound state in the effective potential.