ENG 
ESP Session S24 - Symbolic Computation: Theory, Algorithms and Applications
Tuesday, July 13, 16:00 ~ 16:25 UTC-3
Spectral factorization of algebro-geometric differential operators
Sonia L. Rueda
Universidad Politécnica de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
In 1923, J.L. Burchnall and T.W. Chaundy established a correspondence between commuting differential operators and algebraic curves. It was already known (I. Schur, 1904) that the centralizer of an ordinary differential operator $L$ has a quotient field that is the function field of one variable. Therefore centralizers can be seen as the affine rings of curves, and in a formal sense these are spectral curves.
The theory of commuting differential operators is well developed for commuting ordinary differential operators. However, for rank greater than one (the rank being the greatest common divisor of the orders of all the elements in its centralizer) or for special spectral curves, this theory is not complete enough and continues to evolve. It has broad connections with many branches of modern mathematics, first of all with integrable systems, since explicit examples of commuting operators provide explicit solutions of many non-linear partial differential equations.
Algebro-geometric ordinary differential operators are defined to have nontrivial centralizers. The spectral curve of an algebro-geometric Schrödinger operator $L$ is a plane algebraic curve whose defining equation $f(\lambda,\mu)=0$ can be computed by means of the differential resultant (E. Previato, 1991). The coefficients of $L$ belonging to a differential field $K$, whose field of constants $C$ is algebraically closed and of characteristic zero. In this talk, we present a symbolic algorithm for the factorization of an algebro-geometric Schrödinger operator $L-\lambda$ over the field $K(\Gamma)$ of its spectral curve $\Gamma$, using differential subresultants [1]. This is what we call a "spectral factorization" and our ultimate goal is an effective approach to the direct spectral problem. Since the field of constants $C(\Gamma)$ of $K(\Gamma)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the spectral problem $L\Psi=\lambda\Psi$ over $\Gamma$ called "Spectral Picard-Vessiot field" of $L-\lambda$, defined in [2]. The spectral parameter $\lambda$ is not a free parameter. Restricting to the case of rational spectral curves, we transform the original spectral problem to one-parameter form, where the field of the curve is now $C(\tau)$, with a free parameter $\tau$.
We extended our symbolic algorithm to the factorization of a third order algebro-geometric ordinary differential operator $L-\lambda$ [4]. In this context, the first example of a non-planar spectral curve arises. The spectral factorization over planar spectral curves in the case of algebro-geometric operators of fourth order with rank $2$, see [3], is a complicated problem; we will show how the spectral factorization is affected by the rank of the operator.
References:
[1] J.J. Morales-Ruiz. S.L. Rueda, and M.A. Zurro. Factorization of KdV Schr\" odinger operators using differential subresultants. Adv. Appl. Math., 120:102065, 2020.
[2] J.J. Morales-Ruiz. S.L. Rueda, and M.A. Zurro. Spectral Picard-Vessiot fields for algebro-geometric Schrödinger operators . To appear in Ann. Inst. Fourier. See arXiv:1708.00431v3, 2021.
[3] E. Previato, S.L. Rueda, and M.A. Zurro. Commuting Ordinary Differential Operators and the Dixmier Test. SIGMA Symmetry Integrability Geom. Methods Appl., 15(101):23 pp., 2019.
[4] S.L. Rueda and M.A. Zurro. Factoring Third Order Ordinary Differential Operators over Spectral Curves. See arXiv:2102.04733v1, 2021.
Joint work with Maria-Angeles Zurro (Universidad Autónoma de Madrid), Juan J. Morales-Ruiz (Universidad Politécnica de Madrid) and Emma Previato (Boston University).