  Particle--anti-particle interpretation under spatially inhomogeneous external
fields within the framework of quantum field theory is a nontrivial problem. In
this paper, we focus on the two interpretations established in [Phys. Rev. D
93, 045002 (2016)] and [Prog. Theor. Exp. Phys. 2022, 073B02 (2022)], both of
which give consistent results of vacuum instability and pair production. To
shed light on their differences, a pair production under a potential step
assisted by a weak and oscillating electric field is discussed. It is shown
that the potential step and the oscillating field, each insufficient for vacuum
decay, can produce pairs when combined. In addition, the two pictures give rise
to quantitative differences in the number of created pairs at the second-order
perturbation of the oscillating field. It might provide a clue to investigate
the correct particle--anti-particle interpretation by comparing the result with
numerical simulations or experiments.
