Bispectral Darboux Transformations: The Generalized Airy Case

Details Bispectral Darboux Transformations: The Generalized Airy Case Bispectral Darboux Transformations: The Generalized Airy Case

1996-06-24

arxiv.org/abs/q-alg/9606018v1

Mathematics

Quantum Algebra

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

Scientific paper

10.1016/S0167-2789(96)00208-4

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.

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