Exact scaling relations for the time-dependent Dirac equation (TDDE)

Establish exact scaling relations with respect to the nuclear charge Z for the time-dependent Dirac equation describing hydrogen-like ions in the dipole approximation, analogous to the exact hydrogenic scaling relations known for the time-dependent Schrödinger equation, to enable predictive mapping across species.

Background

In the non-relativistic regime governed by the time-dependent Schrödinger equation (TDSE), hydrogen-like ions exhibit exact scaling relations that map coordinates, time, and frequency (and the field strength for continuous fields) across nuclear charge Z. These mappings allow the absorption spectra of arbitrary hydrogen-like ions to be inferred directly from a hydrogen reference, and they serve as a rigorous benchmark for assessing relativistic effects.

In contrast, within the fully relativistic framework of the time-dependent Dirac equation (TDDE), the paper states that exact scaling relations are unknown. This gap motivates the development and evaluation of semi-relativistic approximations, but a complete, exact scaling theory for the TDDE remains unresolved and would enable direct, symmetry-based predictions for highly charged ions analogous to those available in the TDSE.