  Exotic excitations arise at the interface between a three-dimensional
topological insulator (TI) and superconductors. For example, Majorana fermions
with a linear dispersion, $E\sim k$, exist in a short $\pi$ Josephson junction
on the TI surface. We show that in these systems, the Andreev bound states
spectrum becomes nearly flat at zero energy when the chemical potential is
sufficiently away from the Dirac point. The flat dispersion is well
approximated by $E\sim k^N$, where $N$ scales with the chemical potential.
Similar evolution from linear to flat dispersion also occurs for the subgap
spectrum of a periodic superconducting proximity structure, such as a TI
surface in contact with a stripe superconductor.
