One-dimensional Coulomb problem in Dirac materials

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.