Equivalence of local and separable realizations of discontinuity-inducing contact interaction

We consider the regularized approximations of the discontinuity-inducing si ngular zero-range interaction in one-dimensional quantum mechanics. We prov e that two known approximations, one in terms of local potential and the ot her in terms of separable function are equivalent. We further show that the interaction allows the perturbative treatment through the coupling constan t renormalization.