Reading quantum field theory text, I am confused on regard to whether perturbation (or interaction equivalently) hamiltonian added to free-field hamiltonian is time-dependent or not. In Heisenberg picture, hamiltonian is an operator, so it is obviously time-dependent, but what about in Schrodinger picture? Would hamiltonian still be time-dependent overall and thus for perturbation hamiltonian?

And by QFT, I restrict to physically relevant QFTs.