Quantum Inverse Scattering Method for the $q$-Boson Model and Symmetric Functions

The purpose of this paper is to show that the quantum inverse scattering method for the so-called $q$-boson model has a nice interpretation in terms of the algebra of symmetric functions. In particular, in the case of the phase model (corresponding to $q=0$) the creation operator coincides (modulo a scalar factor) with the operator of multiplication by the generating function of complete homogeneous symmetric functions, and the wave functions are expressed via the Schur functions $s_\lambda(x)$. The general case of the $q$-boson model is related in a similar way to the Hall–Littlewood symmetric functions $P_\lambda(x;q^2)$.