Bispectral Darboux Transformations: The Generalized Airy Case
q-alg
2009-10-30 v1 Quantum Algebra
Exactly Solvable and Integrable Systems
solv-int
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 . As a result, the bispectrality of a large family of algebras of rank 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.
Keywords
Cite
@article{arxiv.q-alg/9606018,
title = {Bispectral Darboux Transformations: The Generalized Airy Case},
author = {Alex Kasman and Mitchell Rothstein},
journal= {arXiv preprint arXiv:q-alg/9606018},
year = {2009}
}
Comments
LaTeX, to appear in Physica D, a nicer postscript version is available at http://www.math.uga.edu/~kasman/