Rational KdV Potentials and Differential Galois Theory

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schr"odinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.