Bispectral Darboux Transformations: The Generalized Airy Case

Abstract

This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order $r>1$. As a result, the bispectrality of a large family of algebras of rank $r$ is demonstrated. An involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters, generalizing Wilson's rank one bispectral involution. Spectral geometry and the relationship to the Sato grassmannian are discussed.

LaTeX, to appear in Physica D, a nicer postscript version is available at http://www.math.uga.edu/~kasman/

References in corpus (1)

• Highest weight modules of W_{1+infty}, Darboux transformations and the bispectral problem

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