[en] A manifestly gauge-invariant, time-dependent perturbation theory for a non-relativistic, spinless charged particle interacting with a classical, external electromagnetic radiation field is formulated with especial reference to the treatment of the degenerate bound states and the states in the continuum. It is found that the gauge-invariant and the conventional perturbative solutions agree on the net transition rates but disagree on the transition amplitudes and transition probabilities. The treatment of a realistic physical system can be obtained by combining the present treatment with that previously obtained for a non-degenerate system. (author)