Schrödinger operators with Coulomb-like potentials

Abstract

We study the convergence of 1D Schrödinger ope\-rators $H_{\epsilon }$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$\alpha {\delta }^{\prime }(x)+\beta \delta (x)+\gamma /|x|\text{or}\alpha {\delta }^{\prime }(x)+\beta \delta (x)+\gamma /x.$$ 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 ${\delta }^{\prime }$-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 $L^{\infty }(\mathbb{R})$-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.

Corrected version 2, 18 pages, 4 figures

References in corpus (3)

• Self-adjoint extensions of Coulomb systems in 1,2 and 3 dimensions
• Schrödinger operators with singular rank-two perturbations and point interactions
• Two-parametric ${\delta }^{\prime }$-interactions: approximation by Schrödinger operators with localized rank-two perturbations

Cited by in corpus (2)

• The bound-state solutions of the one-dimensional hydrogen atom
• Quantum graphs: Coulomb-type potentials and exactly solvable models