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Journal of Mathematical Physics

Quantum inverse scattering method for a nonlinear N-wave resonance interaction system

Huan-Qiang Zhou, Lin-Jie Jiang

Published Mar 1, 1989• 1 citation
Publisher

Abstract

The quantum inverse scattering method is used for the study of a nonlinear N-wave resonance interaction system. The Yang–Baxter relations are solved to get various commutation relations for the scattering data operators. The energy spectrum of the quantum Hamiltonian for the model is determined and the existence of the quantum bound states is analyzed. In the classical limit, the corresponding r matrix is found, and the well-established correspondence between the classical solitons and the quantum bound states is examined. Finally, the integrability of the same model but with both fermion fields and boson fields is discussed.

DOI
10.1063/1.528436

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The quantum inverse scattering method is used for the study of a nonlinear N-wave resonance interaction system. The Yang–Baxter relations are solved to get various commutation relations for the scattering data operators. The energy spectrum of the quantum Hamiltonian for the model is determined and the existence of the quantum bound states is analyzed. In the classical limit, the corresponding r matrix is found, and the well-established correspondence between the classical solitons and the quantum bound states is examined. Finally, the integrability of the same model but with both fermion fields and boson fields is discussed.
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The quantum inverse scattering method relates the Bethe Ansatz to the theory of classical completely integrable differential equations. These are sometimes called soliton equations. The modern way to solve them is called the classical inverse scattering method. In a sense this is a nonlinear generalization of the Fourier transform. In this Part the quantum inverse scattering method is expounded. The main statements of the classical inverse scattering method necessary for quantization are given in Chapter V where the Lax representation is introduced. The Hamiltonian structure of integrable models is also discussed along with the infinite number of integrals of motion. The most convenient method of analyzing the Hamiltonian structure relies on the classical r -matrix. Some concrete models will be considered. Chapter VI is devoted in particular to the quantum inverse scattering method. The R-matrix, which is the main object of this method, is introduced. The Yang-Baxter equation for the R-matrix is discussed. The main statements of the method are given and a number of examples are presented. The algebraic formulation of the Bethe Ansatz, one of the main achievements of the quantum inverse scattering method, is presented in Chapter VII. The notion of the determinant of the transition matrix in the quantum case is introduced in this chapter. (This is closely related to the concept of the antipode in quantum groups.)
Through the quantum inverse scattering method the nonlinear Schrödinger model is studied for the attractive case. Making use of the bound state operators and the quantum Gelfand-Levitan equation, the matrix elements of the field operator φ( x ) and the density operator φ + ( x )φ( x ) are explicitly calculated. Each matrix element is found to be expressed in a product form. Based on the results, the connection between classical soliton and the quantum field theory is investigated. It is shown that the wave form of the classical soliton is related to the matrix element of the field operator in the limit n →∞, where n is the particle number making the bound state. The powerfulness of the quantum Gelfand-Levitan equation is stressed.
The quantum inverse scattering method is applied to the nonlinear Schrodinger model of fermions with attractive interactions. Both scattering and bound-state operators are constructed. Among state operators, there are generators of infinitely many conserved quantities and the creators of the eigenstates of these quantities. The commutation relations and the eigenvalues of physical interest are calculated. The quantum Gelfand-Levitan equations are established. The Fourier transform of the connected part of the 4-point Green function and the two body S matrix are calculated explicitly.
By using a variant of the quantum inverse scattering method, commutation relations between all elements of the quantum monodromy matrix of the bosonic massive Thirring (BMT) model are obtained. Using those relations, the quantum integrability of BMT model is established and the S-matrix of two-body scattering between the corresponding quasiparticles has been obtained. It is observed that for some special values of the coupling constant, there exists an upper bound on the number of quasiparticles that can form a quantum-soliton state of the BMT model. We also calculate the binding energy for a N-soliton state of the quantum BMT model.