Schr\"odinger operators with Coulomb-like potentials

We study the convergence of 1D Schr"odinger ope-rators $H_{\epsilon }$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$\alpha {\delta }^{\mathrm{\prime }}(x)+\beta \delta (x)+\gamma \mathrm{/}\mathrm{\mid }x\mathrm{\mid }\text{or}\alpha {\delta }^{\mathrm{\prime }}(x)+\beta \delta (x)+\gamma \mathrm{/}x\mathrm{.}$$ The limit behaviour of $H_{\epsilon }$ in the norm resolvent topology, as $\epsilon \to 0$, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for ${\delta }^{\mathrm{\prime }}$-like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of $L^{\mathrm{\infty }}(\mathbb{R})$-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.