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The Bispectral Problem
Edited by: John Harnad and Alex Kasman, Centre de Recherches Mathématiques, Université, Montreal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Proceedings & Lecture Notes
1998; 235 pp; softcover
Volume: 14
ISBN-10: 0-8218-0949-0
ISBN-13: 978-0-8218-0949-5
List Price: US$83 Member Price: US$66.40
Order Code: CRMP/14

Although originally posed in the context of mathematical problems related to medical imaging, the bispectral problem is now closely related to other topics and has connections to many areas of pure and applied mathematics. The central theme of this book is the search for solutions to eigenvalue problems that satisfy additional equations in the spectral parameter, for example, pairs of eigenvalue equations. This problem, which looks very simple at first, has turned out to be both deep and difficult. Moreover, this concept of bispectrality has been shown to be useful in many active areas of current research in mathematics and physics.

Following several years of exciting new results on the subject, in March 1997 the Centre de Recherches Mathématiques held the first scientific meeting devoted exclusively to the bispectral problem. Collected in this volume are contributions from the speakers at this meeting. The participants at this workshop included a majority of those researchers who have made significant contributions to the subject and many others working on related problems.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Graduate students, research mathematicians and physicists interested in the bispectral problem.

Part 1. Bispectrality
• B. Bakalov, E. Horozov, and M. Yakimov -- Automorphisms of the Weyl algebra and bispectral operators
• Y. Berest -- Huygens' principle and the bispectral problem
• F. A. Grunbaum -- Some bispectral musings
• L. Haine -- Beyond the classical orthogonal polynomials
• J. Harnad -- Bispectral operators, dual isomondromic deformations and the Riemann-Hilbert dressing method
• A. Kasman -- Darboux transformations and the bispectral problem
• F. Levstein and L. F. Matusevich -- The discrete version of the bispectral problem
• M. Rothstein -- Explicit formulas for the Airy and Bessel bispectral involutions in terms of Calogero-Moser pairs
• V. Spiridonov, L. Vinet, and A. Zhedanov -- Bispectrality and Darboux transformations in the theory of orthogonal polynomials
• A. P. Veselov -- Baker-Akhiezer functions and the bispectral problem in many dimensions
• G. Wilson -- Bispectral algebras of ordinary differential operators
• J. P. Zubelli -- The bispectral problem, rational solutions of the master symmetry flows, and bihamiltonian systems
Part 2. Related Topics
• V. Kac and J. van de Leur -- The geometry of spinors and the multicomponent BKP and DKP hierarchies
• F. Magri -- The Hamiltonian route to Sato Grassmannian
• V. B. Matveev -- Darboux transformations in associative rings and functional-difference equations
• A. Y. Orlov -- Remarks about the Calogero-Moser system and the KP equation
• Subject index