Confinement-induced resonance from the generalized Gross-Pitaevskii equations

The confinement-induced resonances for trapped bosons in the cigar-shaped and pancake geometries are studied within the generalized Gross-Pitaevskii equations, which are a simplified version of the Hartree-Fock-Bogoliubov approximation. Although the Hartree-Fock-Bogoliubov method is considered applicable only for small interparticle interactions, the resonance denominators for the chemical potential are obtained in both quasi-one and quasi-two dimensions. A useful integral representation of the one-particle Green's function are found for the cylindrical confinement. We find the position of a smoothed resonance for the chemical potential in the pancake geometry at positive scattering length.