  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.
