In the time dependent perturbation theory of quantum mechanics, we start off with the assumption that the Hamiltonian of the perturbed system can be written as the sum of the Hamiltonian of the unperturbed system and the perturbation itself. i.e., $$ H = H_0 + W(t) $$ where W(t) is the time dependent perturbation. My query is when is this possible? Is it always true that the Hamiltonian of the perturbed system can be split in this way? If not, can you provide me with an example where splitting the total Hamiltonian is not possible?
The reason I'm asking this question is because, as far as I understand, the perturbations always reveal themselves in the form of time varying potential energy function, in which the time dependency is woven into the potential energy function. How are we separating out the time dependent part out here?