PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.Flow.Stone

Let $T$ be a self-adjoint operator on a Hilbert space $H$ with domain $\mathcal{D}(T)$. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group associated with $T$ via the spectral theorem, defined by the spectral integral $U(t) = \int_{\mathbb{R}} e^{it\lambda} \, d\mu_S(\lambda)$. For any vector $x \in \mathcal{D}(T)$, the limit of the difference quotient as $t \to 0$ converges in the norm topology of $H$ to $iTx$: $$\lim_{t \to 0} \frac{e^{itT}x - x}{t} = iTx$$ where $i$ is the imaginary unit.

Let $T$ be a self-adjoint operator acting on a Hilbert space $H$, and let $\mu_S$ be its associated spectral measure. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group defined by the spectral integral $U(t) = \int_{\mathbb{R}} e^{it\lambda} \, d\mu_S(\lambda)$. For any vector $x$ in the domain of $T$, the function $t \mapsto U(t)x$ is differentiable at $t = 0$, and its derivative is given by $$\left. \frac{d}{dt} (e^{itT}x) \right|_{t=0} = iTx$$ where $i$ is the imaginary unit.

Let $H$ be a Hilbert space and $T$ a self-adjoint operator on $H$ with domain $\text{dom}(T)$. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group associated with $T$ via the spectral theorem, defined by $U(t) = \int_{\mathbb{R}} e^{it\lambda} \, d\mu_S(\lambda)$. For any vector $x \in \text{dom}(T)$ and any time $s \in \mathbb{R}$, the function $t \mapsto U(t)x$ is differentiable at $t = s$ with derivative $$\frac{d}{dt} \left( e^{itT}x \right) \bigg|_{t=s} = e^{isT}(iTx)$$ where $i$ is the imaginary unit.

Let $H$ be a Hilbert space and $T$ be a self-adjoint operator with domain $\mathcal{D}(T)$ and spectral measure $\mu_S$. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group associated with $T$, defined by the spectral integral $U(t) = \int_{\mathbb{R}} e^{it\lambda} \, d\mu_S(\lambda)$. For any vector $x \in H$, $x$ belongs to the domain of $T$ if and only if there exists a vector $y \in H$ such that the limit of the difference quotient converges as $t \to 0$: $$\lim_{t \to 0} \frac{U(t)x - x}{t} = y$$

Let $H$ be a Hilbert space and $T$ be a self-adjoint operator on $H$ with domain $\text{dom}(T)$. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group associated with $T$ via the spectral theorem. For any vector $x \in H$, $x$ belongs to $\text{dom}(T)$ if and only if the vector-valued function $t \mapsto U(t)x$ is differentiable at $t = 0$, i.e., there exists a vector $y \in H$ such that $\frac{d}{dt}(U(t)x)|_{t=0} = y$.

Let $H$ be a Hilbert space and $T$ a self-adjoint operator with associated spectral measure $\mu_S$. Let $U(t) = e^{itT}$ be the strongly continuous one-parameter unitary group defined by the spectral integral $$U(t) = \int_{\mathbb{R}} e^{it\lambda} \, d\mu_S(\lambda)$$ For any $t \in \mathbb{R}$, the adjoint of the unitary operator $U(t)$ satisfies $U(t)^* = U(-t)$.