Bound and resonance states of the nonlinear Schrodinger equation in simple model systems

We study the stationary nonlinear Schrodinger equation, or Gross-Pitaevskii equation, for a single delta potential and a delta-shell potential. These model systems allow analytical solutions, and thus provide useful insight into the features of stationary bound, scattering and resonance states of the nonlinear Schrodinger equation. For the single delta potential, the influence of the potential strength and the nonlinearity is studied as well as the transition from bound to scattering states. Furthermore, the properties of resonance states in a repulsive delta-shell potential are discussed.