Free evolution conserves analyticity but what if the initial state $\Psi_0$ is real-analytic but the acting Hamiltonian includes a non-analytic or even singular potential, like the Coulomb central potential $V(x) = -1/|x|$? One would suppose that in this case, analyticity gets destroyed, at least for almost all times $t \neq 0$.

If one scans the literature a positive analyticity result can be found in ^[1] for analytic potentials in one space dimension. Counter-examples where analyticity is lost when potentials are not that nice are hard to find, there is a statement of yuggib on StackExchange Physics that evolution with a Coulomb potential already kills continuous differentiability, but without reference or proof. And there is Theorem 1.2 in ^[2] that states that the fundamental solution to the time-dependent Schrödinger equation is nowhere $\mathcal{C}^1$ for a certain class of potentials. But this class does not include Coulomb potentials and the properties of the fundamental solution do not necessarily carry over to the solution itself.