  We consider a multiband metal with deep primary bands and a shallow secondary
one. In the normal state the system undergoes Lifshitz transition when the
bottom of the shallow band crosses the Fermi level. In the superconducting
state Cooper pairing in the shallow band is induced by the deep ones. As a
result, the density of electrons in the shallow band remains finite even when
the bottom of the band is above the Fermi level. We study the density of states
in the system and find qualitatively different behaviors on the two sides of
the Lifshitz transition. On one side of the transition the density of states
diverges at the energy equal to the induced gap, whereas on the other side it
vanishes. We argue that this physical picture describes the recently measured
gap structure in shallow bands of iron pnictides and selenides.
