  Strong field ionization is difficult to treat theoretically due to the
simultaneous need to treat bound state dynamics accurately and continuum
dynamics efficiently. We address this problem by decomposing the time dependent
Schroedinger equation (TDSE) into an inhomogeneous form, in which a precomputed
bound state acts as a source term for a time dependent tunneling component. The
resulting theory is equivalent to the full TDSE when exact propagation is used,
and reduces or eliminates a major source of wavefunction error when propagation
is approximated. The gauge invariance of the resulting theory is used to
clarify an apparent gauge dependence which has long been observed in the
context of strong field S-matrix theory.
