In this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian by a pseudo-Hamiltonian PAW via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as and smoother eigenfunctions. In practice, the pseudo-Hamiltonian PAW has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schrödinger operator with double Dirac potentials.