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Journal of Mathematical Analysis and Applications

On the use of Lie-Bäcklund operators in quantum mechanics

ATHANASSIOS S. FOKAS, Paco A. Lagerstrom

Published Apr 1, 1980• 5 citations
Publisher
DOI
10.1016/0022-247x(80)90133-x

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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 BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund 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
Abstract We derive the quantum analogue of a Bäcklund transformation for the quantized Ablowitz–Ladik chain, a space discretization of the nonlinear Schrödinger equation. The quantization of the Ablowitz–Ladik chain leads to the q -boson model. Using a previous construction of Baxter’s Q -operator for this model by the author, a set of functional relations is obtained which matches the relations of a one-variable classical Bäcklund transform to all orders in . We construct also a second Q -operator and show that it is closely related to the inverse of the first. The multi-Bäcklund transforms generated from the Q -operator define the fusion matrices of a 2D TQFT and we derive a linear system for the solution to the quantum Bäcklund relations in terms of the TQFT fusion coefficients.