Generalized Gelfand-Dikii equation and solitonic electric fields for fermionic Schwinger pair production

In previous work on Schwinger pair creation in purely time-dependent fields, it was shown how to construct ``solitonic'' electric fields that do not create scalar pairs with an arbitrary fixed momentum. We show that this construction can be adapted to the fermionic case in two inequivalent ways, both closely related to supersymmetric quantum mechanics for reflectionless potentials, and both leading to the vanishing of the density of created pairs at certain values of the P\"oschl-Teller like index $p$ of the associated Schr\"odinger equation. For one of them, we are able to demonstrate that the pair non-creation can be interpreted as a quantum interference effect using the phase-integral formalism. Asymptotically for large $p$, here scalar particles are not created for integer $p$ and fermions are not created for half integer $p$. Thus for any given momentum we can construct electric fields that create scalar particles but not spinor particles, and vice versa. In the scalar QED case, the solitonic fields had originally been found using the Gelfand-Dikii equation, which is related to the resolvent of the mode equation, and through it to the (generalized) KdV equation [38]. This motivates us to develop for the spinor QED case, too, an evolution equation that can be considered as a fermionic generalization of the Gelfand-Dikii equation.