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Quantum Integrable Systems




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ISBN 9781584883807
Published January 28, 2004 by Chapman and Hall/CRC
424 Pages - 36 B/W Illustrations

 
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Book Description

The study of integrable systems has opened new horizons in classical physics over the past few decades, particularly in the subatomic world. Yet despite the field now having reached a level of maturity, very few books provide an introduction to the field accessible to specialists and nonspecialists alike, and none offer a systematic survey of the more recent advances.

This book presents and clarifies the developments of the last ten years in quantum integrable systems. After a preliminary discussion of the fundamentals of classical nonlinear integrable systems, the authors explore the quantum domain. Their approach emphasizes physical systems and the use of concrete examples, and they take care to establish the relationship between new and older methods. The presentation includes the first comprehensive discussion of the quantum Bäcklund transformation Q-operator and various techniques related to algebraic Bethe Ansatz that are not available elsewhere in book form.

In Quantum Integrable Systems, researchers active in the field have an up-to-date source for recent advances and new techniques, and nonspecialists finally have an accessible introduction to the concepts and basic tools they need to explore and exploit the wide-ranging applicability of the subject.

Table of Contents

NONLINEAR SYSTEMS AND CLASSICAL IST
Introduction
Definition of Integrability
Lax Pair Technique
Inverse Scattering Transform
Hamiltonian Structure
COORDINATE BETHE ANSATZ
Introduction
Nonlinear Systems and the CBA
Fermionic System
Boundary Condition in Bethe Ansatz
Heisenberg Spin Chain
Spin of the Bethe Ansatz State
Other Integrable Models
YANG-BAXTER EQUATION
Introduction
General Description
Factorized Scattering
Baxter's Star Triangle Relation
Vertex Models
Reflection Equation Algebra
CONTINUOUS INTEGRABLE SYSTEMS
Introduction
Quantum Continuous Integrable Systems
Conserved Quantities
Nonultralocal systems and the YBE
Operator Product Expansion and YBE
Finite Boundary Conditions
Modified Classical Yang-Baxter Equation
ALGEBRAIC BETHE ANSATZ
Introduction
Discrete Self Trapping Model
Asymmetric XXZ Model in a Magnetic Field
Analytical Bethe Ansatz
Off-Shell Bethe Ansatz
Nested Bethe Ansatz
Fusion Procedure
Fusion Procedure for Open chains
Fusion Procedure for Transfer Matrices
Application of Fusion Procedure
INTEGRABLE LONG-RANGE MODELS
Introduction
Long-Range Models from the ABA
Symmetry Transformation
Calogero-Moser Models
SEPARATION OF VARIABLES
Introduction
Hamilton-Jacobi Equation
Sklyanin's Method for SoV
Goryachev-Chaplygin Top
Quantum Case and the Role of Lie Algebra
Bi-Hamiltonian Structure and SoV
SoV for GCM Model
SoV and Boundary Conditions
BÄCKLUND TRANSFORMATIONS
Introduction
Permutability Theorem
Bäcklund Transformations and Classical Inverse Scattering
Bäcklund Transformations from Riccati Equation
Darboux Bäcklund Transformations
The Exponential Lattice
Canonical Transformations
Group Property of Bäcklund Transformations
Recent Developments in Bäcklund Transformation Theory
Sklyanin's Formalism for Canonical Bäcklund
Transformations
Extended Phase Space Method
Quantization of Bäcklund Transformations
Method of Projection Operators
QUANTUM GLM EQUATION
Introduction
Quantum GLM Equation
Quantum Floquet Function
Exact Quantization
Quantum GLM Equation in a Continuous System
Bound States and an Alternative Approach
APPENDICES
Direct Product Calculus
Grassman Algebra
Bethe Ansatz Equation
AKNS Problem
BIBLIOGRAPHY
INDEX

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Reviews

"The authors present the recent studies of the quantum inverse scattering method and problems related to it."
-Zentralblatt MATH