Factorization and Integrable Systems

by Gohberg, Israel; Manojlovic, Nenad; Dos Santos, Antonio Ferreira

• ISBN13: 9783764369385
• ISBN10: 3764369388
• Format: Hardcover
• Copyright: 2003-06-01
• Publisher: Birkhauser

Summary

In September 2000 a Summer School on "Factorization and Integrable Systems" was held at the University of Algarve in Portugal. The main aim of the school was to review the modern factorization theory and its application to classical and quantum integrable systems. The program consisted of a number of short courses given by leading experts in the field. The lecture notes of the courses have been specially prepared for publication in this volume. The book consists of four contributions. I. Gohberg, M.A. Kaashoek and I.M. Spitkovsky present an extensive review of the factorization theory of matrix functions relative to a curve, with emphasis on the developments of the last 20-25 years. The group-theoretical approach to classical integrable systems is reviewed by M.A. Semenov-Tian-Shansky. P.P. Kulish surveyed the quantum inverse scattering method using the isotropic Heisenberg spin chain as the main example.

Table of Contents

Preface	vii
An Overview of Matrix Factorization Theory and Operator Applications
I. Gohberg
M.A. Kaashoek
I.M. Spitkovsky
Introduction	1	(2)
General theorems	3	(21)
Factorization in decomposing Banach algebras	24	(14)
Generalized Lp factorization	38	(19)
State space method	57	(46)
References	95	(8)
Matrix Riemann-Hilbert Problems Related to Branched Coverings of CP1
Dmitry Korotkin
Introduction	103	(4)
Riemann-Hilbert problem with quasi-permutation monodromies and algebraic curves	107	(4)
Riemann surfaces. Rauch variational formulas	111	(5)
Solution of Riemann-Hilbert problems with quasi-permutation monodromies and Szego kernel	116	(3)
Isomonodromic tau-function and Cauchy-Riemann determinants	119	(12)
Acknowledgements	127	(1)
References	127	(4)
Quantum Groups and Integrable Models
Peter P. Kulish
Introduction	131	(2)
Algebraic Bethe Ansatz and QISM	133	(11)
Yang-Baxter equation	144	(4)
Thermodynamic limits	148	(4)
Conclusion	152	(3)
Acknowledgements	152	(1)
References	153	(2)
Integrable Systems and Factorization Problems
M.A. Semenov-Tian-Shansky
Introduction	155	(3)
A few preliminaries: Poisson brackets, coadjoint orbits, etc.	158	(4)
Classical r-matrices and Lax equations	162	(5)
Classical Yang-Baxter identity	167	(3)
A finite-dimensional example	170	(4)
Loop algebras and the Riemann problem	174	(5)
More examples	179	(5)
Zero curvature equations	184	(18)
Difference equations and Poisson-Lie groups	202	(17)
Acknowledgements	217	(1)
References	217	(2)
Programme of the Summer School on Factorization and Integrable Systems	219