We present an extension of the resolvent-operator method (ROM), originally designed for atomic systems, to extract differential photoelectron spectra (in photoelectron-and nuclear-kinetic energy) for diatomic molecules interacting with strong, ultrashort laser fields in the single active electron approximation. The method is applied to the study of H 2 + photodissociation and photoionization by femtosecond laser pulses in the XUV-IR frequency range. In particular, the method is tested (i) in the perturbative regime, for few-photon absorption and bound-bound electronic transitions, and (ii) in the strong-field regime, in which multiphoton absorption and tunneling are present. In the latter case, we show how the differential ROM allows one to track the transition between both regimes. We also analyze isotopic effects by comparing the dynamics of H 2 + and D 2 + ionization for different pulses.