Schrödinger operators with Coulomb-like potentials

Abstract

We study the convergence of 1D Schrödinger ope\-rators $H_\varepsilon$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$ αδ'(x)+βδ(x)+γ/|x|\quad\text{or}\quad αδ'(x)+βδ(x)+γ/x. $$ The limit behaviour of $H_\varepsilon$ in the norm resolvent topology, as $\varepsilon\to 0$, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for $δ'$-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 $L^\infty(\mathbb{R})$-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.