#### Conference abstracts

Session B3 - Symbolic Analysis

July 14, 15:00 ~ 15:25

## Factorization of stationary Schrodinger operators over KdV spectral curves

### Sonia Rueda

These results will allow us to give new explanations to some well known results related with the celebrated KdV hierarchy. We will describe the centralizer of a Schrödinger operator $L=-\partial^2+u$, for stationary potentials $u$ subject to constrains given by the KdV hierarchy, using results of Goodearl. We present an algorithm to factor Schrödinger operators $L_s-\lambda=-\partial^2+u_s-\lambda$ with $u_s$ satisfying the KdV${}_s$ equation. Previous results, for hyperelliptic curves, construct factorizations as formulas using $\theta$-functions. The method we are presenting is effective and it points out the fact that closed formulas for factors of the Schrödinger operator over the curve can be obtained using: a global parametrization of the spectral curve (if it exists), and the subresultant formula operator. Furthermore, one can extend the coefficient field of $L_s$ to the field of a spectral curve and then to the Liouvillian extension given by $\Psi'=\phi_s\Psi$, with $\phi_s$ satisfying the Ricatti equation $\phi'+\phi^2=u_s-\lambda$. In this manner, we will describe the Picard-Vessiot fields of $L_s-\lambda$. A spezialization process in $\lambda$ can be adapted to our methods.