Rational KdV Potentials and Differential Galois Theory

Abstract

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schrödinger equation for rational time-dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated with the Schrödinger 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.