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1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials

For real bounded functions Φand Ψof compact support, we prove the norm resolvent convergence, as εand νtend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{ε, ν}= -D^2+αε^{-2}Φ(ε^{-1}x)+βν^{-1}Ψ(ν^{-1}x), provided the ratio ν/εhas a finite or infinity limit. The limit operator S_0 depends on the shape of Φand Ψas well as on the limit of ratio ν/ε. If the potential αΦpossesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.