In a sufficiently strong electrostatic step, the positive-energy incident channel overlaps a negative-kinetic-energy channel on the far side. For a charged scalar field the outgoing antiparticle branch has negative charge current, allowing reflected particle current larger than the incident current while signed charge is conserved. The external source supplies energy for particle-antiparticle creation. This requires a multiparticle interpretation; negative transmitted charge current is not a negative probability.
For the outgoing negative-energy branch , matching the field and its spatial derivative gives these signed charge-current ratios. Their sum is one although reflection exceeds one. The ideal sharp step has a singular stationary matching denominator at in the supercritical range. It is not a finite single-particle probability prediction; wavepackets, source preparation and field quantization matter for its physical interpretation.
Set and continue both roots from the upper half-plane. The right-hand wavenumber is positive above the positive-energy threshold, positive imaginary inside the gap, and negative below the negative-energy threshold. Its group velocity in the last region remains rightward because both momentum and kinetic energy are negative. Choosing a positive real root there instead specifies an incoming wavepacket from the right.
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