**CRM Proceedings & Lecture Notes**

Volume: 14;
1998;
235 pp;
Softcover

MSC: Primary 33; 34;
Secondary 58; 13; 35; 70

Print ISBN: 978-0-8218-0949-5

Product Code: CRMP/14

List Price: $83.00

Individual Member Price: $66.40

**Electronic ISBN: 978-1-4704-3928-6
Product Code: CRMP/14.E**

List Price: $83.00

Individual Member Price: $66.40

# The Bispectral Problem

Share this page *Edited by *
*John Harnad; Alex Kasman*

A co-publication of the AMS and Centre de Recherches Mathématiques

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.

#### Table of Contents

# Table of Contents

## The Bispectral Problem

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Part 1. Bispectrality 110
- Automorphisms of the Weyl algebra and bispectral operators 312
- Huygens’ principle and the bispectral problem 1120
- Some bispectral musings 3140
- Beyond the classical orthogonal polynomials 4756
- Bispectral operators, dual isomondromic deformations and the Riemann-Hilbert dressing method 6776
- Darboux transformations and the bispectral problem 8190
- The discrete version of the bispectral problem 93102
- Explicit formulas for the Airy and Bessel bispectral involutions in terms of Calogero-Moser pairs 105114
- Bispectrality and Darboux transformations in the theory of orthogonal polynomials 111120
- Baker-Akhiezer functions and the bispectral problem in many dimensions 123132
- Bispectral algebras of ordinary differential operators 131140
- The bispectral problem, rational solutions of the master symmetry flows, and bihamiltonian systems 139148
- Part 2. Related Topics 157166
- The geometry of spinors and the multicomponent BKP and DKP hierarchies 159168
- The Hamiltonian route to Sato Grassmannian 203212
- Darboux transformations in associative rings and functional-difference equations 211220
- Remarks about the Calogero-Moser system and the KP equation 227236
- Subject index 235244
- Back Cover Back Cover1245

#### Readership

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