# Schr\"odinger operators with Coulomb-like potentials

## TL;DR

This paper investigates how one-dimensional Schrödinger operators with Coulomb-like potentials behave under regularization, revealing their limit interactions and constructing approximations for Coulomb potentials with different boundary behaviors.

## Contribution

It characterizes the norm resolvent limits of Schrödinger operators with Coulomb-like potentials and constructs explicit $L^
abla( ext{R})$-approximations for these potentials.

## Key findings

- Limits depend on regularization method and zero-energy resonances.
- All possible limit point interactions are classified.
- Constructed explicit $L^
abla( ext{R})$-approximations for Coulomb potentials.

## Abstract

We study the convergence of 1D Schr\"odinger ope\-rators $H_\varepsilon$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$ \alpha \delta'(x)+\beta \delta(x)+\gamma/|x|\quad\text{or}\quad \alpha \delta'(x)+\beta \delta(x)+\gamma/x. $$ 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 $\delta'$-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.