  We suggest a better mathematical method, fractional calculus, for studying
the behavior of the atom-field interaction in photonic crystals. By studying
the spontaneous emission of an atom in a photonic crystal with one-band
isotropic model, we found that the long-time inducing memory of the spontaneous
emission is a fractional phenomenon. This behavior could be well described by
the fractional calculus. And the results show no steady photon-atom bound state
for the atomic resonant transition frequency lying in the proximity of allowed
band edge which is encountered in the previous study [J. Opt. B: Quantum
Semiclass. Opt. {\bf 5}, R43 (2003)]. The correctness of this result is
validated by the ``cut-off smoothing'' density of photon states (DOS) with
fractional calculus. By obtaining a rigorous solution without the
multiple-valued problem for the system, we show the method of fractional
calculus has logically concise property.
