Rational KdV Potentials and Differential Galois Theory

Juan J. Morales-Ruiz; Mar\'ia-\'Angeles Zurro; Raquel S\'anchez-Cauce; Sonia Jim\'enez

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Rational KdV Potentials and Differential Galois Theory

Sonia Jiménez, Juan J. Morales-Ruiz, Raquel Sánchez-Cauce, María-Ángeles Zurro

Mathematical Physics · Submitted 2 Aug 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

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arXiv ID1808.00743 (version 3)

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CitationJuan J. Morales-Ruiz, Mar\'ia-\'Angeles Zurro, Raquel S\'anchez-Cauce, Sonia Jim\'enez. "Rational KdV Potentials and Differential Galois Theory." arXiv:1808.00743 (2018). https://pith.science/paper/A6QKDOWI

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In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schr\"odinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials.	contribution 0.99	0.92	0.11	asserted 0.97
s1	In particular, we compute the fundamental matrices of the linear systems associated to the Schr\"odinger equation.	result 0.67	0.95	0.08	asserted 1.00
s2	Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.	result 0.99	0.94	0.54	asserted 1.00

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Original abstract

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schr\"odinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 3 sentences without a confident label

Cites as background 4 papers

For a spectral study of the Picard–Vessiot extensions see [ref] .

Spectral Picard-Vessiot fields for Algebro-geometric Schr\"odinger operators · Case $E ≠0$ · 0.95
In some articles, such as [ref] and [ref] , the authors studied the general Schrödinger equation, i.e., the potential $u$ is a differential indeterminate which satisfies KdV$_1$ equation.

Poisson groups and differential Galois theory of Schroedinger equation on the circle · Introduction · 0.92
Brezhnev in three papers [ref] also consider the Galois groups associated to spectral problem for some KdV potentials.

Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials · Introduction · 0.96

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