According to the Schrödinger equation
$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$
the transformation $U_t:\mathcal H\to \mathcal H: \Psi\mapsto \Psi(t)=e^{-itH}\Psi$ for every $t\in\mathbb R$ is a unitary transformation of the Hilbert space $\mathcal H$, and since from this equation,
$$U_tU_s=U_{t+s}\tag 2,$$
the mapping, $t\mapsto U_t$ it is a one-parameter subgroup of the unitary group.
Starting from the other direction, if we suppose that the evolution of the states is a one-parameter subgroup of the unitary group, Stone's theorem yields (1).
But, since quantum mechanical states are not elements of $\mathcal H$, but they are elements of the projective Hilbert space $\mathcal P(\mathcal H)$, time evolution isn't a one-parameter subgroup of the unitary group, but it is a one-parameter subgroup of the projective unitary group, that is, by Wigner's theorem, instead of (2), only
$$U_tU_s=\omega(t,s)U_{t+s}\tag 3$$
must hold, where $\omega(t,s)$ is a complex number of modulus $1$ depending on $t$ and $s$.
From (2), Stone's theorem yields (1), but from (3), what theorem yields what?
Best Answer
There are two theorems relevant here. The former proves that actually, under natural hypotheses, the multipliers $\omega$ can be removed and one can safely apply the Stone theorem. This theorem is an immediate corollary of the Bargmann theorem as the Lie group $\mathbb{R}$ is (simply connected and) Abelian and its Lie algebra cohomology is trivial, or also can be directly proved (it was established by Wigner independently).
The latter shows that, if the representation is unitary, continuity is almost automatic, since non Borel-measurable functions are really difficult to find.