Bispectral property, Darboux transformation and the Grassmannian Gr^{rat}

LIBERATI, JOSE I.

LETTERS IN MATHEMATICAL PHYSICS

SPRINGER

Año: 1997 vol. 41 p. 321 - 332

0377-9017

We give a complete description of the differential operators of the commutative algebra associated with elements of the Grassmannians Gr^rat and Grrat introduced by G. Wilson, in terms of Darboux transformations. This extends some earlier results of Duistermaat and Grünbaum about the bispectral problem, i.e.: find the differential operators L(x,∂x) such that there exists a family of eigenfunctions ψ(x,z) which also satisfies a differential equation of the form B(z,∂z)ψ=t(x)ψ. We give a new proof of one of Wilson´s results, an explicit formula for the operator B in terms of the τ-functions, and we extend the characterization of the functions Θ given by Zubelli and Wright.