Bispectral Darboux transformations: The generalized Airy case

✍ Scribed by Alex Kasman; Mitchell Rothstein

• Book ID: 104297099
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 837 KB
• Volume: 102
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/s0167-2789(96)00208-4

✦ Synopsis

This paper considers Darboux transformations ofa 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.