Nonlinear Schr\"odinger equations with exceptional potentials

We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of modified scattering for this model, that is, linear scattering modulated by a phase. Our approach is based on the spectral theorem for the perturbed linear Schr\"odinger operator and a factorization technique, that allows us to control the resonant nonlinear term. We make some parity assumptions in order to control the small-energy behavior of the scattering coefficients and of the wave functions.