In this talk I will discuss recent work with Jonas Luhrmann and Wilhem Schlag on the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one, under equivariant symmetry. Among the main results are the determination of the spectrum of the (non-selfadjoint) linearized operator, uncovering a remarkable L^2 growth phenomenon related to zero-energy resonance, and a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.