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

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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 Sε, ν= -D²+αε⁻²Φ(ε⁻¹x)+βν⁻¹Ψ(ν⁻¹x), provided the ratio ν/εhas a finite or infinity limit. The limit operator S₀ 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 S₀ describes a non trivial point interaction at the origin. Otherwise S₀ is the direct sum of the Dirichlet half-line Schroedinger operators.