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Journal of the Physical Society of Japan

The Quantum Nonlinear Schrödinger Model; Conserved Quantities

Miki Wadati, Atsuo Kuniba

Published Jan 15, 1986• 12 citations
Publisher

Abstract

By means of the quantum inverse scattering method, conserved quantities for the quantum nonlinear Schrödinger model are studied. It is shown that the series expansion of the scattering data operator A (λ) provides not only explicit forms of the conserved quantities but also key informations for the open problems related to the classical and semiclassical theories.

DOI
10.1143/jpsj.55.76

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By means of the quantum inverse scattering method, conserved quantities for the quantum nonlinear Schrödinger model are studied. It is shown that the series expansion of the scattering data operator A (λ) provides not only explicit forms of the conserved quantities but also key informations for the open problems related to the classical and semiclassical theories.
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 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.
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 version of the infinite set of polynomial conserved quantities for the nonlinear Schr\"odinger equation in 1+1 dimensions is discussed. In the conventional formulation of the theory, which describes a quantum many-particle system with a \ensuremath{\delta}-function potential, it is shown that an infinite set of quantum polynomial commuting operators does not exist. However, by adopting a rather unconventional quantization method, we construct an infinite set of quantum polynomial commuting operators explicitly in a parallel way with the Korteweg--de Vries and the modified Korteweg--de Vries equation cases discussed in a previous paper. Based on these results a new scheme of solvable quantum field theories is proposed.
A quantum nonlinear Schr?dinger model with the most general supermatrices, having structure (n-k,k)(m-l,l), is studied by using the quantum inverse scattering method. The eigenstates of the Hamiltonian and the infinite number of the conserved quantities of the system are discussed, In particular, the N-particle bound states with the mixture of bosons and fermions are found. The energy of the N-particle eigenstate are ∑j=1Nλj2 and Np2-(c2/12)N(N2-1) for the scattering stateand the bound state respectively.