Quantum inverse scattering problem for models with $GL(N)$-invariant $R$-matrix

We consider quantum models solvable by the algebraic Bethe ansatz. Physical states of such the models are polynomials in the entries of the quantum monodromy matrix acting on a vacuum vector. The quantum monodromy matrix is a non-local operator, and hence, the physical states also are non-local. We develop a method to calculate matrix elements of local operators in the basis of the physical states. It turns out that they are expressed in terms of the form factors of the monodromy matrix. Thus, we solve the quantum inverse scattering problem in a weak sense.