Departamento de Matemáticas UAM

Prelectura de Tesis

Raquel Sánchez Cauce

Fecha y hora: 24 de septiembre de 2018 a las 11:00h

Lugar: Aula 520, modulo 17

Título: Differential Galois Theory for some Spectral Problems
Conferenciante: Raquel Sánchez Cauce

Directores de tesis: Juan José Morales Ruiz (Dpto. de Matemática Aplicada. Universidad Politécnica de Madrid) y  María Ángeles Zurro Moro (Dpto. de Matemáticas. Universidad Autónoma de Madrid)

Resumen: In this talk we will introduce the Picard-Vessiot Theory for integrable systems and the Darboux transformations. First, we will present our results on the differential Galois groups for Ablowitz-Kaup-Newell-Segur systems, which are an important kind of integrable systems depending on a spectral parameter $lambda$.

Next we will focus on  the Schrödinger equation $(-partial^{^2}+u)psi=-lambda^{^2} psi$ associated to the Korteweg de Vries hierarchy (KdV hierarchy for short). We will show the algebraic structure of the  fundamental matrices for the Schrödinger equation with potential $u$ in a fixed family of KdV rational potentials. As a by product, we will obtain the differential Galois groups  associated with the mentioned spectral problem. We will also compute  non trivial examples in the $1+1$ dimensional case using SAGE.

Moreover, we will establish the deep relationship between the singularities of the spectral curves, the Darboux transformations and the fundamental matrices for the KdV hierarchy.

Secondly, we will present a family of rational complex potentials $u$ depending on a parameter. We will show that these functions are KdV potentials and compute fundamental matrices for the corresponding Schrödinger equation.

Finally, we will use Darboux transformations for studying orthogonal differential systems from a galoisian point of view. Here the techniques of tensor products of  differential systems are essential tools. Explicit formulas for these matrix Darboux transformations are computed using Maple.