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Prelectura de tesis

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

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.
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.