  Here two theoretical models of semi-classical matrix and quantum Green's
function are developed for the system of x-ray thin-film planar cavity with
inner-shell electronic resonances. The semi-classical model is based on the
matrix formalism to treat each layer as the propagating matrix. The crucial
idea is to expand the propagating matrix of the resonant atomic layer under
ultrathin-film approximation, then derive the analytical expression of the
spectral observation, i.e, the cavity reflectance. The typical cavity effects
of cavity enhanced decay rate, cavity induced energy shift and the Fano
interference which were observed in the recent experiments could be
phenomenologically interpreted. The second quantum model employs the analytical
Green's function to solve the cavity system. The system Hamiltonian and the
effective energy-level are derived. The effective energy-level scheme indicates
that the cavity effect acts on the regulation of the intermediated core-hole
state. To test the validity of the semi-classical matrix and quantum Green's
function models, the classical Parratt's formalism and the dispersion
correction of the atomic refractive index are also recalled. Very good
agreements in reflectivity spectra between semi-classical and quantum models
with the Parratt's results are observed. The present semi-classical matrix and
quantum Green's function models will be useful to predict the new phenomena and
optimize the cavity structures for future experiments and promote the emerging
of quantum optical effects with modern x-ray spectroscopy techniques.
