Conference abstracts

Session B3 - Symbolic Analysis

July 14, 15:00 ~ 15:25

Factorization of stationary Schrodinger operators over KdV spectral curves

Sonia Rueda

Universidad Politécnica de Madrid, Spain   -

In 1928, J. L. Burchall and T. W. Chaundy established a correspondence between commuting differential operators and algebraic curves. With the discovery of solitons and the integrability of the Korteweg de Vries (KdV) equation, using the inverse spectral methods, their theory found applications to the study of partial differential equations called integrable (or with solitonic type solutions: Sine-Gordon, non linear Schrödinger, etc). Burchall and Chaundy had discovered the spectral curve, which was later computed by E. Previato, using differential resultants. The spectral curve allows an algebraic approach to handling the inverse spectral problem for the finite-gap operators, with the spectral data being encoded in the spectral curve and an associated line bundle.

In this work, we explore the benefits of using differential resultants to compute the Burchall and Chaundy polynomials. We review the definition of the differential resultant of two ordinary differential operators and its main properties. We revisit Enma Previato's result about the computation of the spectral curve of two commuting differential operators using differential resultants. We use these results to establish the appropriate fields were commuting operators have a common factor, which can be computed using differential subresultants.

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.

Joint work with M. A. Zurro (Universidad Politécnica de Madrid) and J. J. Morales-Ruiz (Universidad Politécnica de Madrid).

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FoCM 2017, based on a nodethirtythree design.