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For a real-valued function V from the Faddeev-Marchenko class, we prove thenorm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon ofone-dimensional Schr\"odinger operators on the line of the form S_\epsilon:=-D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family ofpotentials converges in the sense of distributions to the first derivative ofthe Dirac delta-function, and then the limit of S_\epsilon might be consideredas a "physically motivated" interpretation of the one-dimensional Schr\"odingeroperator with potential \delta'.

Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials