Is the Hamiltonian in the Exercise Truly Time-Dependent?

[aaaa202]

Consider the attached exercise. I am having some trouble understanding exactly what time dependent hamiltonian it refers to. Because from the equation it refers to it seems that the hamiltonian is by definition time independent. Am I to assume that the H diagonal is a time independent hamiltonian which is perturbed by a time dependent potential V(t) or am I to assume that it wants the time evolution in the Heissenberg picture where for an operator:
A(t) = exp(iHt)Aexp(-iHt) (1)
Because then I can just expand the operators a_dagger, a in time and recover (1). But that would hold also for a non diagonalized hamiltonian.
Can anyone explain to me the difference between H(t) and H in this case and why it is not always just given by the formula I am to prove for a diagonalized H - it seems intuitive for me that the time evolution of H should follow the time evolution of a_dagger and a.

 

[BruceW]

I think this problem is just meant to show how you can go through the calculation to explicitly show that H is not time dependent in this case. And you are not meant to assume $a_\nu(t) = \exp(-i\varepsilon_\nu t) a_\nu$ from the start. (By the way, I'm guessing the stuff under the red line is the solution to the problem, right?)