In this paper, we consider the continuous Schrödinger operators \(H_c\) in \(L^2({{\mathbb {R}}})\) with \(\delta \) -point interactions supported on a slightly perturbed lattice \(\Gamma \) of \(\ell {\mathbb {Z}}\) with a small parameter \(\ell >0\) and its discrete approximation. Unlike equidistant lattices \(\ell {{\mathbb {Z}}}\) , it can be natural to construct the one-dimensional discrete Schrödinger operators \(H_d\) in \(\ell ^2(\Gamma )\) by taking inspiration from the Shortley–Welley method utilized in the numerical analysis. In spirit to the method constructed by Exner et al. (Lett Math Phys 112(4):Paper No. 83, 15, 2022) for the equilateral lattice quantum graph Hamiltonian, we give an estimate for the resolvent difference between \(H_c\) and \(H_d\) with an identification operator as \(\ell \rightarrow 0\) .
