PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Existence.GeneratorInvariance

Let $H$ be a Hilbert space and $U$ be a one-parameter unitary group. For any vector $\psi \in H$ and any $t \in \mathbb{R}$, the function $s \mapsto U(t)(U(s)\psi)$ is equal to the function $s \mapsto U(s)(U(t)\psi)$. This expresses the fact that the operators $U(t)$ and $U(s)$ commute for all $t, s \in \mathbb{R}$.

Let $H$ be a Hilbert space and $U$ be a strongly continuous unitary group. For any vectors $\psi, \varphi \in H$ and any time $t \in \mathbb{R}$, if the orbit $s \mapsto U(s)\psi$ is differentiable at $s=0$ with derivative $\varphi$, then the orbit of the translated vector $U(t)\psi$, given by $s \mapsto U(s)(U(t)\psi)$, is also differentiable at $s=0$ with derivative $U(t)\varphi$.

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. For any vector $\psi \in H$, if the orbit $s \mapsto U(s)\psi$ is differentiable at $s = 0$, then for any $t \in \mathbb{R}$, the orbit of the translated vector $U(t)\psi$, given by $s \mapsto U(s)(U(t)\psi)$, is also differentiable at $s = 0$. This implies that the domain of the candidate generator of the unitary group is invariant under the action of the group $U(t)$.

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $\mathcal{D}$ be the candidate domain of the generator of $U$, defined as the $\mathbb{C}$-submodule of $H$ consisting of vectors $\psi$ such that the orbit $s \mapsto U(s)\psi$ is differentiable at $s = 0$. For any vector $\psi \in \mathcal{D}$ and any $t \in \mathbb{R}$, the translated vector $U(t)\psi$ is also an element of $\mathcal{D}$.

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $T$ be the candidate Stone generator with domain $\mathcal{D} \subseteq H$ (consisting of vectors $\psi$ where the orbit $t \mapsto U(t)\psi$ is differentiable at $t=0$). For any vector $\psi \in \mathcal{D}$ and any $t \in \mathbb{R}$, the vector $U(t)\psi$ is also in $\mathcal{D}$, and the generator $T$ commutes with the group action such that $T(U(t)\psi) = U(t)(T\psi)$.

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $D(T)$ be the candidate domain of the generator, consisting of vectors $\psi \in H$ such that the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$, and let $T$ be the candidate Stone generator. For any vector $\psi \in D(T)$ and any time $s \in \mathbb{R}$, the orbit map $r \mapsto U(r)\psi$ is differentiable at $r = s$ with derivative $i U(s) T\psi$, where $i$ is the imaginary unit.