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

Gelfand-Levitan method for operator fields

Dennis B. Creamer, H. B. Thacker, David Wilkinson

Published Mar 15, 1980• 87 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.21.1523

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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.
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 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.