arXiv subjects
Publications and source records attributed to N. Slavnov.
We obtain a determinant representation of normalized scalar products of on-shell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solvability and solve it in terms of determinants of explicitly known matrices.
This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.
The foundation for the theory of correlation functions of exactly solvable models is determinant representation. Determinant representation permit to describe correlation functions by classical completely integrable differential equations [Barough, McCoy, Wu]. In this paper we show that determinant represents works not only for free fermionic models. We obtained determinant representation for the correlation function $<ψ(0,0)ψ^\dagger(x,t)>$ of the quantum nonlinear Schrödinger equation, out of free fermionic point. In the forthcoming publications we shall derive completely integrable equation and asymptotic for the quantum correlation function of this model of interacting fermions.
Correlation functions of exactly solvable models can be described by differential equation [Barough, McCoy, Wu]. In this paper we show that for non free fermionic case differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes time and temperature dependent correlation function $<ψ(0,0)ψ^\dagger(x,t)>_T$ of penetrable Bose gas. The integro-differential equation turns out be the continuum generalization of classical nonlinear Schrödinger equation.