  In this work, we investigate the bound state problem in one dimensional
spin-1 Dirac Hamiltonian with a flat band. It is found that, the flat band has
significant effects on the bound states. For example, for Dirac delta potential
$g\delta(x)$, there exists one bound state for both positive and negative
potential strength $g$. Furthermore, when the potential is weak, the bound
state energy is proportional to the potential strength $g$. For square well
potential, the flat band results in the existence of infinite bound states for
arbitrarily weak potential. In addition, when the bound state energy is very
near the flat band, the energy displays hydrogen atom-like spectrum, i.e., the
bound state energies are inversely proportional to the square of natural number
$n$ (e.g., $E_n\propto 1/n^2, n=1,2,3,...$). Most of the above nontrivial
behaviors can be attributed to the infinitely large density of states of flat
band and its ensuing $1/z$ singularity of Green function. The combination of a
short-ranged potential and flat band provides a new possibility to get infinite
number of bound states and hydrogen atom-like energy spectrum. In addition, our
findings would provide some useful insights in the understanding of many-body
physics of flat band.
