Constraining transmission and reflection probabilities by using the Miller-Good transformation

Abstract

Transmission through and reflection from a potential barrier, and the very closely related issue of particle production from a parametric resonance, are topics of considerable general interest in quantum physics. We have developed a rather general bound on quantum transmission probabilities, and recently applied it to bounding the greybody factors of a Schwarzschild black hole. In this current paper, we take a different tack -- we report a way of using the Miller-Good transformation (which maps an initial Schrodinger equation to a final Schrodinger equation for a different potential) to significantly generalize the previous bound. We then apply this general formalism in a very specific manner to derive a rigorous bound that is "as close as possible" to the usual WKB estimate for barrier penetration.

13 pages, no figures, to appear in the proceedings of the 2nd Regional Conference on Applied and Engineering Mathematics (Penang), Malaysia