Differential equations with respect to a coupling constant: An approach to Čerenkov and stimulated radiations

The PDECC (partial differential equations with respect to coupling constants) formalism is adapted to derive two, physically equivalent, $S$ matrices for calculating the power spectrum of stimulated radiation semiclassically, and the power spectrum of Čerenkov radiation either semiclassically or quantum mechanically (to the lowest order in the perturbation theory). The stimulated radiation resulting from the interaction between electrons and an incident electromagnetic field (a laser, for example) could have frequencies significantly upshifted from those of the incident electromagnetic field. This suggests a possible practical way of generating radiation at higher frequencies than currently available from lasers, as well as submillimeter radiation from microwaves.