Rational KdV Potentials and Differential Galois Theory

TL;DR

This paper applies differential Galois theory to analyze the spectral problem of the Schrödinger equation with rational time-dependent KdV potentials, revealing invariance properties of Galois groups.

Contribution

It introduces a novel application of differential Galois theory to rational KdV potentials, computing fundamental matrices and proving Galois group invariance.

Findings

Computed fundamental matrices for the Schrödinger equation with KdV potentials

Proved invariance of Galois groups under time evolution and Darboux transformations

Established stability of Galois groups for generic spectral parameters

Abstract

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.