Factorization of KdV Schrödinger operators using differential subresultants

We address the classical factorization problem of a one dimensional Schr\"odinger operator \(-\partial^2+u-\lambda\), for a stationary potential \(u\) of the KdV hierarchy but, in this occasion, a "parameter" \(\lambda\). Inspired by the more effective approach of Gesztesy and Holden...

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Bibliographic Details
Published inarXiv.org
Main Authors Morales-Ruiz, Juan J, Rueda, Sonia L, Maria-Angeles Zurro
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 13.02.2019
Subjects
Algorithms
Factorization
Operators (mathematics)
Resultants
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Summary:We address the classical factorization problem of a one dimensional Schr\"odinger operator \(-\partial^2+u-\lambda\), for a stationary potential \(u\) of the KdV hierarchy but, in this occasion, a "parameter" \(\lambda\). Inspired by the more effective approach of Gesztesy and Holden to the "direct" spectral problem, we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make our method fully effective, we 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 Schr\"odinger operators with solitonic potentials.
ISSN:2331-8422