  We develop some new analytic bounds on transmission probabilities (and the
related reflection probabilities and Bogoliubov coefficients) for generic
one-dimensional scattering problems. To do so we rewrite the Schrodinger
equation for some complicated potential whose properties we are trying to
investigate in terms of some simpler potential whose properties are assumed
known, plus a (possibly large) "shift" in the potential. Doing so permits us to
extract considerable useful information without having to exactly solve the
full scattering problem.
