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Journal of Physics A Mathematical and General

Quantum inverse scattering method for multicomponent non-linear Schrodinger model of bosons or fermions with repulsive coupling

Fu‐Cho Pu, Yi-Zhong Wu, Bao‐Heng Zhao

Published Apr 1, 1987• 18 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/5/026

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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.
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.
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).
We consider the nonlinear Schrödinger model of spin-21\frac{2}{1}12​ particles with attractive coupling to show a way to complete an infinite volume formulation of the quantum inverse scattering method. We use not any lattice models but only properties of the Jost functions satisfying given asymptotic boundary conditions in infinities. We find orthonormality relations of scattering data operators as well as underlying ones of the Jost functions and use them to derive operator identities satisfied by inverse operators of scattering data operators. Consequently we can simplyfy constructing the Gel'fand-Levitan equations. We also realize scattering data operators in terms of reflection coefficient operators.
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.
The quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schr\"odinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.