Factorization of KdV Schrödinger operators using differential subresultants (Q2197907)

This paper addresses the effective factorization of the Schrödinger operator \(L-\lambda=-\partial^2+u-\lambda\) for a stationary potential \(u\) in a complex variable, say \(x\), and \(\lambda\) a parameter over the field of coefficients. More precisely the authors address the classical factorization problem of a one dimensional Schrödinger operator \(-\partial^2+u-\lambda\), for a stationary potential \(u\) of the Korteweg-de Vries (KdV) hierarchy but, in this occasion, a parameter \(\lambda\) is considered. Inspired by the more effective approach of Gesztesy and Holden to the direct spectral problem in [\textit{F. Gesztesy} and \textit{H. Holden}, Soliton equations and their algebro-geometric solutions. I: (1+1)-dimensional continuous models. Cambridge: Cambridge University Press (2003; Zbl 1061.37056)], the authors give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization, that in addition allows one parameter form factorizations by means of suitable global parametrizations of the spectral curve. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make their method fully effective, they design a symbolic algorithm to compute the integration constants of the KdV hierarchy, in the case of KdV potentials that become rational under a Hamiltonian change of variable. Explicit computations are carried for Schrödinger operators with solitonic potentials. This paper is organized as follows: Section 1 is an introduction to the subject and Section 2 is devoted to some notations. In Section 3, the authors construct the KdV hierarchy and define differential sub-resultants, reviewing its main properties. Section 4 contains their algorithm for computation of the integration constants of the KdV hierarchy. The authors determine the essential odd order operator \(A_{2s+1}\) of the centralizer of \(L\), that together with \(L\) generates the centralizer as a \(\mathbb{C}\)-algebra \(\mathbb{C}[L, A_{2s+1}]\). Then in Section 5 they describe the centralizer of \(L\) and compute the operator \(A_{2s+1}\). They are ready to review Previato's Theorem, applying it to the computation of the spectral curve of the Lax pair \(\{L, A_{2s+1}\}\). Section 6 deals with factors of KdV Schrödinger operators over spectral curve. Section 7 is devoted to Schrödinger operators for KdV solitons and computed examples.