Wave function collapses and 1/n energy spectrum induced by a Coulomb potential in a one-dimensional flat band system

We investigate the bound state problem in a one-dimensional flat band system with a Coulomb potential. It is found that, in the presence of a Coulomb potential of type I (with three equal diagonal elements), similarly to that in the two-dimensional case, the flat band could not survive. At the same time, the flat band states are transformed into localized states with a logarithmic singularity near the center position. In addition, the wave function near the origin would collapse for an arbitrarily weak Coulomb potential. Due to the wave function collapses, the eigen-energies for a shifted Coulomb potential depend sensitively on the cut-off parameter. For a Coulomb potential of type II, there exist infinite bound states that are generated from the flat band. Furthermore, when the bound state energy is very near the flat band, the energy is inversely proportional to the natural number, e.g.,$E_n\propto$ 1/n, n=1,2,3,... It is expected that the 1/n energy spectrum could be observed experimentally in the near future.