  Motivated by recent experimental advances in ultracold atoms, we analyze a
non-Hermitian (NH) BCS Hamiltonian with a complex-valued interaction arising
from inelastic scattering between fermions. We develop a mean-field theory to
obtain a NH gap equation for order parameters, which are different from the
standard BCS ones due to the inequivalence of left and right eigenstates in the
NH physics. We find unconventional phase transitions unique to NH systems:
superfluidity shows reentrant behavior with increasing dissipation, as a
consequence of non-diagonalizable exceptional points, lines, and surfaces in
the quasiparticle Hamiltonian for weak attractive interactions. For strong
attractive interactions, the superfluid gap never collapses but is enhanced by
dissipation due to an interplay between the BCS-BEC crossover and the quantum
Zeno effect. Our results lay the groundwork for studies of fermionic
superfluidity subject to inelastic collisions.
