Quantum nonlinear Schroedinger equation: Two solutions

The quantum nonlinear Schroedinger equation (QNLS) has attracted much attention recently as a simplest exactly soluble nonlinear model of the quantum field theory in 1+1 space-time. There are two approaches in the literature to solution of QNLS. One is a quantization prescription for the solution of classical nonlinear Schroedinger equation by the inverse scattering method. It is called the quantum inverse method (QIM). The other is an elaboration of the Bethe Ansatz technique. It is called the method of intertwining operators (MIO). The two approaches produce formally different expansions for the QNLS field and for certain Fock space operators associated with it. In this work we give a comparative exposition of both methods. We then show that the QIM expansions and the MIO expansions define the same operators on Fock space. (orig.)