I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are unbounded operators on some Hilbert space $\mathcal{H}$.
Essentially the formula is as follows. Let $\mathcal{H}$ be a Hilbert space and let $A$, $B$, be two self-adjoint unbounded operators with domains $\mathcal{D}(A)$ and $\mathcal{D}(B)$. Then, $$[A, e^{-itB}] = e^{-itB} (e^{itB} A e^{-itB} - A) = -i \int_0^t e^{-i(t-s)B} [A, B] e^{-isB} ds.$$ I suspect one also needs, at the very least, the fact $e^{-itB} \mathcal{D}(A) \subset \mathcal{D}(A)$ (so that both sides of the equation are well-defined and the unbounded commutator on the right-hand side can be defined in the standard way via a sesquilinear form).
Perhaps this has something to do with the map $t \mapsto e^{itB} A e^{-itB}$ being strongly $C^1$ on $\mathcal{D}(A) \cap \mathcal{D}(B)$?
