  A rigorous theoretical framework is developed for a generalized local frame
transformation theory (GLFT). The GLFT is applicable to the following systems:
to Rydberg atoms or molecules in an electric field, or to negative ions in any
combination of electric and/or magnetic fields. A first test application to the
photoionization spectra of Rydberg atoms in an external electric field
demonstrates dramatic improvement over the first version of the local frame
transformation theory developed initially by Fano and Harmin. This revised GLFT
theory yields non-trivial corrections because it now includes the full on-shell
Hilbert space without adopting the truncations in the original theory.
Comparisons of the semi-analytical GLFT Stark spectra with {\it ab initio}
numerical simulations yields errors in the range of a few tens of MHz, an
improvement over the original Fano-Harmin theory whose errors are 10-100 times
larger. Our analysis provides a systematic pathway to precisely describe the
corresponding photoabsorption spectra that should be accurate enough to meet
most modern experimental standards.
