  Modal analysis based on the quasi-normal modes (QNM), also called resonant
states, has emerged as a promising way for modeling the resonant interaction of
light with open optical cavities. However, the fields associated with QNM in
open photonic cavities diverge far away from the scatterer and the possibility
of expanding the scattered field with resonant contributions only has not been
established. Here, we address these two issues while restricting our study to
the case of a dispersionless spherical scatterer. First, we derive the rigorous
pole expansion of the $T$-matrix coefficients that link the scattered to the
incident fields associated with an optical resonator. This expansion evinces
the existence of a non-resonant term. Second, in the time domain, the causality
principle allows us to solve the problem of divergence and to derive a modal
expansion of the scattered field that does not diverge far from the scatterer.
