Introduction The quantum inverse scattering method (QSIM) appears as the quantized form of the classical inverse scattering method. It allows us to reproduce the results of the Bethe Ansatz and to move ahead. QISM is now a well developed branch of mathematical physics. In this chapter the fundamentals of QISM are given and illustrated by concrete examples. In section 1 the general scheme of QISM, which allows the calculation of commutation relations between elements of the transfer matrix (necessary to construct eigenfunctions of the Hamiltonian in Chapter VII) is presented, and the quantum R-matrix is introduced. As in the classical case, the existence of an R-matrix and trace identities ensures that a Lax representation for the model exists. Thus, there are infinitely many conservation laws. The Yang-Baxter equation, which is satisfied by the R-matrix, is discussed in section 2. Some important features of the R-matrix are also mentioned. The trace identities for the quantum nonlinear Schrödinger equation are proved in section 3. The general scheme of QISM is applied to the quantum sine-Gordon and Zhiber-Shabat-Mikhailov models in section 4. Spin models of quantum statistical physics are discussed in section 5. It is shown that a fundamental spin model can be constructed with the help of any given R-matrix. The connection between classical statistical models on a two-dimensional lattice and QISM is established in section 6. QISM is the generalization of the classical inverse scattering method.Introduction The quantum inverse scattering method (QSIM) appears as the quantized form of the classical inverse scattering method. It allows us to reproduce the results of the Bethe Ansatz and to move ahead. QISM is now a well developed branch of mathematical physics. In this chapter the fundamentals of QISM are given and illustrated by concrete examples. In section 1 the general scheme of QISM, which allows the calculation of commutation relations between elements of the transfer matrix (necessary to construct eigenfunctions of the Hamiltonian in Chapter VII) is presented, and the quantum R-matrix is introduced. As in the classical case, the existence of an R-matrix and trace identities ensures that a Lax representation for the model exists. Thus, there are infinitely many conservation laws. The Yang-Baxter equation, which is satisfied by the R-matrix, is discussed in section 2. Some important features of the R-matrix are also mentioned. The trace identities for the quantum nonlinear Schrödinger equation are proved in section 3. The general scheme of QISM is applied to the quantum sine-Gordon and Zhiber-Shabat-Mikhailov models in section 4. Spin models of quantum statistical physics are discussed in section 5. It is shown that a fundamental spin model can be constructed with the help of any given R-matrix. The connection between classical statistical models on a two-dimensional lattice and QISM is established in section 6. QISM is the generalization of the classical inverse scattering method.

Cambridge University Press eBooksAug 19, 199348 citations

Boundary K-matrix for the quantum Mikhailov-Shabat model

J.D. Kim

( We present complete solutions of $K$ -matrix for the quantum Mikhailov-Shabat model. It has been known that there are three diagonal solutions with no free parameters, one being trivial identity solution, the others non-trivial. The most general solutions which we found consist of three families corresponding to each diagonal solutions. One family of solutions depends on two arbitrary parameters. If one of the parameters vanishes, the other must also vanish so that the solutions reduces to trivial identity solution. The other two families for each non-trivial diagonal solutions have only one arbitrary parameter.)( We present complete solutions of $K$-matrix for the quantum Mikhailov-Shabat model. It has been known that there are three diagonal solutions with no free parameters, one being trivial identity solution, the others non-trivial. The most general solutions which we found consist of three families corresponding to each diagonal solutions. One family of solutions depends on two arbitrary parameters. If one of the parameters vanishes, the other must also vanish so that the solutions reduces to trivial identity solution. The other two families for each non-trivial diagonal solutions have only one arbitrary parameter.)

arXiv:hep-th/9412192Dec 21, 199416 citations

( We present complete solutions of

K

-matrix for the quantum Mikhailov-Shabat model. It has been known that there are three diagonal solutions with no free parameters, one being trivial identity solution, the others non-trivial. The most general solutions which we found consist of three families corresponding to each diagonal solutions. One family of solutions depends on two arbitrary parameters. If one of the parameters vanishes, the other must also vanish so that the solutions reduces to trivial identity solution. The other two families for each non-trivial diagonal solutions have only one arbitrary parameter.)