PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.Existence.StoneReconstruction

Given a strongly continuous unitary group $U(t)$ on a Hilbert space $H$, let $T$ be the candidate generator associated with it. This definition constructs the spectral measure $\mu$ on $\mathbb{R}$ associated with the self-adjoint closure $\bar{T}$ of the operator $T$ via the Cayley transform. This spectral measure is represented as a Weak Operator Topology (WOT) spectral measure.

Let $\mathcal{H}$ be a complex Hilbert space and $U(t)$ be a strongly continuous unitary group on $\mathcal{H}$. Let $T$ be the candidate generator associated with $U(t)$. Then the closure of this operator, $\bar{T}$, satisfies the spectral theorem for self-adjoint operators with respect to the spectral measure $\mu$ constructed via the Cayley transform. This provides the spectral representation $\bar{T} = \int_{\mathbb{R}} \lambda \, d\mu(\lambda)$ and identifies the domain of $\bar{T}$ accordingly.

For a given strongly continuous unitary group $U(t)$ and its candidate generator $T$, let $\bar{T}$ be the self-adjoint closure of $T$ with its associated spectral measure $\mu$. This definition assigns to each $t \in \mathbb{R}$ the continuous linear operator $V(t): H \to H$ defined by the spectral integral: $$V(t) = e^{it\bar{T}} = \int_{\mathbb{R}} e^{it\lambda} \, d\mu(\lambda)$$ This represents the one-parameter unitary group reconstructed from the spectral data of the generator's closure.

Let $\mathcal{H}$ be a complex Hilbert space and $T$ be a symmetric, partially defined linear operator on $\mathcal{H}$ with domain $D(T)$. Suppose $y, z: \mathbb{R} \to \mathcal{H}$ are differentiable curves such that for all $t \in \mathbb{R}$, the values $y(t)$ and $z(t)$ belong to $D(T)$. If both curves satisfy the Schrödinger-like differential equation $$\frac{d}{dt} w(t) = i T w(t)$$ for all $t$, and they share the same initial value $y(0) = z(0)$, then $y(t) = z(t)$ for all $t \in \mathbb{R}$.

Let $H$ be a complex Hilbert space and $U(t)$ be a strongly continuous one-parameter unitary group on $H$. Let $T$ be the candidate Stone generator of $U(t)$ with domain $D(T)$. Let $V(t) = e^{it\bar{T}}$ be the reconstructed unitary group obtained via the spectral theorem from the closure $\bar{T}$ of $T$. For any vector $x \in D(T)$ and any time $t \in \mathbb{R}$, the action of the original unitary group $U(t)$ on $x$ coincides with the action of the reconstructed unitary group $V(t)$ on $x$: $$U(t)x = V(t)x$$

Let $H$ be a Hilbert space and $U(t)$ be a strongly continuous one-parameter unitary group on $H$. The domain of the candidate generator $T$ (the operator constructed to satisfy the infinitesimal relationship with $U(t)$ in the proof of Stone's theorem) is dense in $H$.

Let $H$ be a Hilbert space and $U(t)$ be a strongly continuous one-parameter unitary group on $H$. Let $T$ be the associated candidate generator and $\bar{T}$ be its essentially self-adjoint closure. For any vector $x \in H$ and any time $t \in \mathbb{R}$, the unitary group $U(t)$ is recovered by the spectral integral of its generator's closure: $$U(t)x = e^{it\bar{T}}x$$ where $e^{it\bar{T}} = \int_{\mathbb{R}} e^{it\lambda} \, d\mu(\lambda)$ is the unitary operator defined via the spectral measure $\mu$ associated with $\bar{T}$.