One-dimensional Coulomb problem in Dirac materials

TL;DR

This paper provides exact solutions for the one-dimensional Coulomb problem in Dirac materials, revealing critical bandgaps and atomic collapse phenomena in quasirelativistic systems.

Contribution

It derives exact wavefunctions and energy spectra for the Coulomb problem in Dirac-Weyl materials, highlighting the impact of bandgap size on quantum states.

Findings

Existence of critical bandgaps affecting low-lying states

Wavefunctions expressed via Whittaker functions

Atomic collapse phenomena observed

Abstract

We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated Coulomb problems, with the wavefunctions expressed in terms of special functions (namely Whittaker functions), whilst the energy spectrum must be determined via solutions to transcendental equations. Most notably, there are critical bandgaps below which certain low-lying quantum states are missing in a manifestation of atomic collapse.