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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum Gelfand-Levitan equations for nonlinear Schrödinger model of spin-½ particles

Fu‐Cho Pu, Bao‐Heng Zhao

Published Nov 15, 1984• 38 citations
Publisher

Abstract

The quantum Gelfand-Levitan equations for the nonlinear Schr\"odinger model of spin-\textonehalf{} particles are obtained. Two Izergin-Korepin relations are used in the derivation. A new type commutation relation of $L$ operators is introduced to get the commutation relations which are needed for the study of $S$ matrices and Green's functions. As examples, the four-point Green's functions and the two-body $S$ matrices are given.

DOI
10.1103/physrevd.30.2253

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The quantum Gelfand-Levitan equations for the nonlinear Schr"odinger model of spin-\textonehalf{} particles are obtained. Two Izergin-Korepin relations are used in the derivation. A new type commutation relation of LLL operators is introduced to get the commutation relations which are needed for the study of SSS matrices and Green's functions. As examples, the four-point Green's functions and the two-body SSS matrices are given.
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 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.
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.
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.
A formalism for treating the many-body problem of composite particles is presented for the case of composite particles consisting of two fermions. Commutation relations for composite-particle operators are derived, as well as a sum rule satisfied by the composite-particle Green's function. In an approximation that shuts off the interactions between composite particles in a consistent manner, the dynamical equation for the one-composite-particle Green's function is solved and the distribution function for the composite particles obtained. Possible applications to real systems are discussed.