Bispectral property, Darboux transformation and the Grassmannian \(Gr^{rat}\)
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.
scientific article; zbMATH DE number 1059643