  The evolution of inflationary fluctuations can be recast as an inverse
scattering problem. In this context, we employ the Gel'fand-Levitan method from
inverse-scattering theory to reconstruct the evolution of both the inflaton
field freeze-out horizon and the Hubble parameter during inflation. We
demonstrate this reconstruction procedure numerically for a scenario of
slow-roll inflation, as well as for a scenario which temporarily departs from
slow-roll. The field freeze-out horizon is reconstructed from the accessible
primordial scalar power spectrum alone, while the reconstruction of the Hubble
parameter requires additional information from the tensor power spectrum. We
briefly discuss the application of this technique to more realistic cases
incorporating estimates of the primordial power spectra over limited ranges of
scales and with specified uncertainties.
