ABSTRACT

We have so far discussed several features of quantum integrable systems, many of which are similar to the classical ones, while others are valid only for quantum systems. The Bethe ansatz together with its variations all belong to the latter category. As mentioned earlier, the Bethe ansatz allows us to calculate the spectrum of the excitations when the nonlinear system is quantized. However, there is a basic difference between the quantum inverse scattering method formulated by means of the algebraic Bethe ansatz and classical inverse scattering transform. While in the classical inverse scattering transform, one determines or reconstructs the form of the nonlinear field as a function of (x, t), in the quantum inverse scattering method one can expect to compute the “excitation levels” only, but not the “shape” of the nonlinear object. In other words, one cannot compute or reconstruct the analog of the classical field. In classical inverse scattering, the reconstruction of the fields is done with the aid of the well-known Gelfand-Levitan-Marchenko (GLM) equation.