  We investigate the onset of superconductivity in magnetic field for a clean
two-dimensional multiple-band superconductor in the vicinity of the Lifshitz
transition when one of the bands is very shallow. Due to small number of
carriers in this band, the quasiclassical Werthamer-Helfand approximation
breaks down and Landau quantization has to be taken into account. We found that
the transition temperature TC2(H) has giant oscillations and is resonantly
enhanced at the magnetic fields corresponding to full occupancy of the Landau
levels in the shallow band. This enhancement is especially pronounced for the
lowest Landau level. As a consequence, the reentrant superconducting regions in
the temperature-field phase diagram emerge at low temperatures near the
magnetic fields at which the chemical potential matches the Landau levels. The
specific behavior depends on the relative strength of the intraband and
interband pairing interactions and the reentrance is most pronounced in the
purely interband coupling scenario. The reentrant behavior is suppressed by the
Zeeman spin splitting in the shallow band, the separated regions disappear
already for very small spin-splitting factors. On the other hand, the
reentrance is restored in the resonance cases when the spin-splitting energy
exactly matches the separation between the Landau levels. The predicted
behavior may realize in the gate-tuned FeSe monolayer.
