Gabor products and a phase space approach to nonlinear analysis

We introduce and study continuity properties of the Gabor product g and relate it to the well-known product formula for the short-time Fourier transform (STFT). We derive a phase space representation of the cubic nonlinear Schrodinger equation in terms of the Gabor product, and discuss how the Gabor product can be used in the study of nonlinear dynamics of mixed quantum states.