Bispectral property, Darboux transformation and the Grassmannian \(Gr^{rat}\)
type: Item
instance of: scholarly article
title: Bispectral property, Darboux transformation and the Grassmannian \(Gr^{rat}\)
zbMATH Open document ID: 0893.34075
DOI: 10.1023/A:1007389326917
published in: Letters in Mathematical Physics
publication date: 1998-02-17T00:00:00Z
review text: An ordinary differential operator \(L(x,\partial_x)\) is said to be bispectral if there is a family of eigenfunctions \(\psi(x,z)\) depending on the parameter \(z\), i.e. \(L\psi(x,z)=f(z)\psi(x,z)\), where \(f\) is a nonconstant function, which satisfies a differential equation in the spectral parameter, i.e. there is a differential operator \(B(z,\partial_z)\) such that \(B\psi(x,z)=\Theta(x)\psi(x,z)\) for some nonconstant function \(\Theta\). Different approaches to characterize bispectral operators have been used by \textit{J. J. Duistermaat} and \textit{F. A. Grünbaum} [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)] on the one side and \textit{G. Wilson} [J. Reine Angew. Math. 442, 177-204 (1993; Zbl 0781.34051)] and Wright on the other side, using certain Grassmannians. In the present paper, these two methods are compared, and results of Duistermaat and Grünbaum are reproved by using Wilson's method. Also, explicit formulas for \(B\) are given.
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1059643
zbMATH Keywords: bispectral property
zbMATH Keywords: Darboux transformation
zbMATH Keywords: Grassmannian
zbMATH Keywords: KdV hierarchy
author: Jose I. Liberati
MaRDI profile type: Publication
full work available at URL: a:1007389326917
OpenAlex ID: W1532078710
description: scientific article; zbMATH DE number 1059643