Generalized Gelfand-Dikii equation 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 on a generalization of that equation for spinor QED. We discuss a number of solvable special cases from this point of view. In previous work, two of the authors had shown for the scalar case how to use the well-known solitonic solutions of the KdV equation to construct P\"oschl-Teller like electric fields that do not pair create at some fixed but arbitrary momentum. Here, we present numerical evidence that this construction can be adapted to the fermionic case by a mere change of parameters.