Abstract
We give a complete description of the differential operators of the commutative algebra associated with elements of the Grassmannians Grrat 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.
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Liberati, J.I. Bispectral Property, Darboux Transformation and the Grassmannian Grrat . Letters in Mathematical Physics 41, 321–332 (1997). https://doi.org/10.1023/A:1007389326917
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DOI: https://doi.org/10.1023/A:1007389326917