For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formUnder certain conditions the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and then the limit S 0 of S ε might be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ ′ .