Generality of the Schrödinger equation

Question

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?

Answer 1

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).

1. (Wigner-Bargmann) If $\mathbb{R}\ni t \mapsto U_t$ is a projective-unitary (so there are your $\omega$) representation such that $|\langle \psi|U_t\phi\rangle|$ is a continuous function of $t$ for every choice of $\psi,\phi$ in the Hilbert space, then it is possible to re-arrange the multipliers $\omega$ (by multiplying the $U_t$ with suitable phases $U’_t:= e^{if(t)}U_t$) in order to obtain a strongly-continuos properly unitary representation.

The latter shows that, if the representation is unitary, continuity is almost automatic, since non Borel-measurable functions are really difficult to find.

2. (von Neumann) if $\mathbb{R} \ni t \mapsto \langle \psi| U_t\phi\rangle$ is Borel measurable for every choice of $\psi,\phi$ of a separable Hilbert space where each $U_t$ acts as a unitary operator and the $U_t$ form a group representation of $\mathbb{R}$, then $t\to U_t$ is strongly continuous. So we can apply the Stone theorem.