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Journal of Physics A Mathematical and General

Algebraic Bethe ansatz for the nonlinear Schrodinger model. I. Multicomponent fields

Yu-Sen Zhou

Published May 21, 1988• 10 citations
Publisher

Abstract

The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).

DOI
10.1088/0305-4470/21/10/016

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For pt.I see ibid., vol.21, p.2391, (1988). The Bethe ansatz equations for a mixture of species of fermions and bosons in one dimension, interacting with a repulsive delta potential, are obtained. It is shown that the matrix periodic boundary conditions of the system can be obtained from those of a fermion or boson system. Finally, the extension of these results to the nonlinear Schrodinger model with a supermatrix is made.

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The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).
The generalisation of the quantum inverse scattering method to the study of direct and inverse problems for the multicomponent non-linear Schrodinger model of bosons or fermions with repulsive coupling is made. Two sets of Yang-Baxter equations are solved to obtain commutation relations between the scattering state operators. The eigenfunctions have been constructed for the infinite number of conserved quantities and the eigenvalues of the first three conserved quantities-number of particles, momentum and energy-are obtained. The global Izergin-Korepin relations and the relations between the quantum Jost functions are derived, and from them, the quantum Gel'fand-Levitan equations are established. Finally, the series expansion for the field operators in terms of the scattering state operators is written out explicitly.
We have reformulated the Bethe Ansatz approach to a Derivative Nonlinear Schrödinger equation, known to be a non-ultralocal system. In this present version we have shown that an approach exactly similar to the usual NLS case is possible. Furthermore the Yang-Baxter equation and factorization is shown to be valid. Afterwards the R (μ, v ) matrix has been constructed from the Bethe Ansatz result (which is impossible otherwise, as the Faddeev approach is not applicable) and the commutation rules for the scattering data is deduced. These are then utilised to construct the algebraic Bethe ansatz eigen states. Next the periodicity condition is invoked and the eigenvalues are determined via some algebraic equations which are also seen to be reproduced via the algebraic approach. Lastly we consider the possibility of exceptional states in the algebraic Bethe states.
Abstract In this work we present a general construction of integrable models describing boson tunneling in multi-well systems. We show how the models may be derived through the quantum inverse scattering method and solved by algebraic Bethe ansatz means. From the transfer matrix we find only two conserved operators. However, we construct additional conserved operators through an alternative approach. As a consequence the models admit multiple pseudovacua, each associated with a set of Bethe ansatz equations (BAEs). We show that all sets of BAEs are needed to obtain a complete set of eigenstates.
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.