I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$ and $y(0)=y_0 \in D(H_0)$ where $V$ is not a bounded operator, but an unbounded one, $u \in C([0,T])$, and $H_0$ self-adjoint.
If $V$ is bounded, then one would obtain a solution preserving the domain of $H_0$ basically by Duhamel's formula. That situation is really standard.
However, I am not so sure what can be said in the case of unbounded $V.$ There seem to be a few really unpleasant domain problems that arise in this case. Obviously, $V$ cannot be completely arbitrary, but should maybe satisfy something like relative boundedness with respect to $H_0.$ At least, this would be my first guess.
Is there a paper or a book that discusses this situation in detail? I suppose since this equation has been studied a lot that this theory is well-developed.
