The thermal oxidation of the graphite leads to the removal of monoatomic carbon layers from the surface and formation of circular pits on the exposed plane. Near the pit edges the scanning tunneling spectroscopy measurements show a series of very narrow flat regions on the I/V characteristics. The observed I/V flat regions appear only when the characteristic is recorded very close to the pit edges. The appearance of tunneling current steps can be explained by the oscillating character of the local density of states near the pit edges. A simple quantum mechanical model in 2D based on boundary condition for an electron wave function is proposed.