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

Schwinger pair creation in a purely time-dependent electric field can be reduced to an effective quantum mechanical problem using a variety of formalisms. Here we develop an approach based on the Gelfand-Dikii equation for scalar QED, and extent it to spinor QED. We discuss some solvable special cases from this point of view. It was previously shown how to use the well-known solitonic solutions of the KdV equation 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 leading to the vanishing of the pair-creation rate at certain values of the P\"oschl-Teller like index $p$ of the associated Schr\"odinger equation. Thus for any given momentum we can construct electric fields that create scalar particles but not spinor particles, and also the other way round. Therefore, while often spin is even neglected in Schwinger pair creation, in such cases it becomes decisive.