424 Pages 36 B/W Illustrations
    by Chapman & Hall

    424 Pages
    by Chapman & Hall

    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.

    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

    Biography

    Roy Chowdhury, Asesh

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