Journal of the Physical Society of Japan

Correspondence between QISM and Exchange Operator Formalism

Hideaki Ujino, Miki Wadati

Published Nov 15, 1995• 7 citations

Abstract

It is shown that the quantum inverse scattering method (QISM) and the exchange operator formalism, which are independently developed to deal with the quantum Calogero-Moser-Sutherland models, give exactly the same operator algebra in the space of wave functions of identical particles.

DOI: 10.1143/jpsj.64.4121

Related papers

The Quantum Inverse Scattering Method

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

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

Integrability of the Quantum Calogero-Moser Model

Hideaki Ujino, Kazuhiro Hikami, Miki Wadati

The integrability of the quantum Calogero-Moser model is proved through the quantum inverse scattering method. By use of the Lax operator L , a new formula for the conserved operators is found. The formula suggests that Tr L n , which are conserved in the classical case, are not necessarily conserved operators. The Hénon-type conserved operators are also presented.

Journal of the Physical Society of JapanOct 15, 199250 citations

Various Approaches to Spectral Problems for Integrable Systems in the QISM

Igor V. Komarov

Algebraic Bethe Ansatz, separation of variables and Baxter's method of functional equation are three main approaches to finding spectrum of commuting integrals of motion in the frame of quantum inverse scattering method. Their connections are discussed.

International Journal of Modern Physics AJan 10, 19972 citations

Quantum Current Algebra Symmetries and Integrable Many-Particle Schrödinger Type Quantum Hamiltonian Operators

Dominik Prorok, Anatolij K. Prykarpatski

Based on the G. Goldin’s quantum current algebra symmetry representation theory, have succeeded in explaining a hidden relationship between the quantum many-particle Hamiltonian operators, defined in the Fock space, their factorized structure and integrability. Interesting for applications quantum oscillatory Hamiltonian operators are considered, the quantum symmetries of the integrable quantum Calogero-Sutherland model are analyzed in detail.

SymmetryAug 1, 20191 citation

A New Solution of the Supersymmetric TJ Model by Means of the Quantum Inverse Scattering Method

Fabian H. L. Eßler, V. E. Korepin

We construct the enveloping fundamental spin model of the t-J hamiltonian using the Quantum Inverse Scattering Method (QISM), and present all three possible Algebraic Bethe Ansätze. Two of the solutions have been previously obtained in the framework of Coordinate Space Bethe Ansatz by Sutherland and by Schlottmann and Lai, whereas the third solution is new. The formulation of the model in terms of the QISM enables us to derive explicit expressions for higher conservation laws.

arXiv:hep-th/9207007Jul 3, 199211 citations

Exact coherent states in one-dimensional quantum many-body systems with inverse-square interactions

Dae-Yup Song

For the models of N-body identical harmonic oscillators interacting through potentials of homogeneous degree -2, the unitary operator that transforms a system of time-dependent parameters into that of unit spring constant and unit mass of different time scale is found. If the interactions can be written in terms of the differences between positions of two particles, it is also shown that the Schr\"odinger equation is invariant under a unitary transformation. These unitary relations can be used not only in finding coherent states from the given stationary states in a system, but also in finding exact wave functions of the Hamiltonian systems of time-dependent parameters from those of time-independent Hamiltonian systems. Both operators are invariant under the exchange of any pair of particles. The transformations are explicitly applied for some of the Calogero-Sutherland models to find exact coherent states.

arXiv:quant-ph/0102022Physical Review AFeb 8, 20018 citations