  We apply Milne's phase-amplitude representation [W. E. Milne, Phys. Rev. 35,
863 (1930)] to a scattering problem involving disjoint classically allowed
regions separated by a barrier. Specifically, we develop a formalism employing
different sets of amplitude and phase functions --- each set of solutions
optimized for a separate region --- and we use these locally adapted solutions
to obtain the true value of the scattering phase shift and accurate tunneling
rates for ultra-narrow shape resonances. We show results for an illustrative
example of an attractive potential with a large centrifugal barrier.

Atomic Physics