  We investigate the $t$-$W$ scheme for the anti-ferromagnetic XXX spin chain
under both periodic and open boundary conditions. We propose a new
parametrization of the eigenvalues of transfer matrix. Based on it, we obtain
the exact solution of the system. By analyzing the distribution of zero roots
at the ground state, we obtain the explicit expressions of the eigenfunctions
of the transfer matrix and the associated $\mathbb{W}$ operator (see (2.8) and
(3.20)) in the thermodynamic limit. We find that the ratio of the quantum
determinant with the eigenvalue of $\mathbb{W}$ operator for the ground state
exhibits exponential decay behavior. Thus this fact ensures that the so-called
inversion relation (the $t-W$ relation without the $W$-term) can be used to
study the ground state properties of quantum integrable systems with/without
$U(1)$-symmetry in the thermodynamic limit.
