PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.Existence.CandidateGenerator

A vector $\psi$ in a Hilbert space $H$ satisfies this predicate if the map $t \mapsto U(t)\psi$ from $\mathbb{R}$ to $H$ is differentiable at $t = 0$, where $U$ is a one-parameter unitary group.

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a one-parameter unitary group. Let $\psi_1, \psi_2 \in H$ be two vectors. If the orbit maps $t \mapsto U(t)\psi_1$ and $t \mapsto U(t)\psi_2$ are differentiable at $t = 0$, then the orbit map of their sum, $t \mapsto U(t)(\psi_1 + \psi_2)$, is also differentiable at $t = 0$.

Let $U$ be a one-parameter unitary group on a Hilbert space $H$. If a vector $\psi \in H$ is such that the map $t \mapsto U(t)\psi$ is differentiable at $t = 0$, then for any scalar $c \in \mathbb{C}$, the map $t \mapsto U(t)(c\psi)$ is also differentiable at $t = 0$.

Let $U$ be a strongly continuous one-parameter unitary group on a Hilbert space $H$. The zero vector $0 \in H$ is differentiable at $t=0$ along its orbit under $U$; that is, the map $t \mapsto U(t)0$ is differentiable at $t = 0$.

Given a one-parameter unitary group $U$ on a Hilbert space $H$, the candidate domain of the Stone generator is the $\mathbb{C}$-submodule of $H$ consisting of all vectors $\psi$ for which the map $t \mapsto U(t)\psi$ is differentiable at $t = 0$.

Given a vector $\psi$ in the candidate domain of the Stone generator for a one-parameter unitary group $U(t)$, this function returns the derivative of the orbit $t \mapsto U(t)\psi$ at $t = 0$. That is, for $\psi \in \text{stoneCandidateDomain}$, the function returns the vector $v = \left. \frac{d}{dt} U(t)\psi \right|_{t=0} \in H$.

Let $U(t)$ be a one-parameter unitary group on a Hilbert space $H$. For any vector $\psi$ in the candidate domain of the Stone generator (the submodule of vectors for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t=0$), the map $t \mapsto U(t)\psi$ has a derivative at $t=0$ equal to the vector $\left. \frac{d}{dt} U(t)\psi \right|_{t=0}$ (formally defined as `stoneCandidateDeriv`).

Let $U(t)$ be a one-parameter unitary group on a Hilbert space $H$. Let $\text{stoneCandidateDomain}$ be the submodule of vectors $\psi \in H$ such that the orbit $t \mapsto U(t)\psi$ is differentiable at $t=0$. For any vectors $\psi_1, \psi_2 \in \text{stoneCandidateDomain}$, the derivative of the orbit of their sum at $t=0$ is the sum of their individual derivatives at $t=0$: $$\left. \frac{d}{dt} U(t)(\psi_1 + \psi_2) \right|_{t=0} = \left. \frac{d}{dt} U(t)\psi_1 \right|_{t=0} + \left. \frac{d}{dt} U(t)\psi_2 \right|_{t=0}$$

Let $U(t)$ be a one-parameter unitary group on a Hilbert space $H$. Let $\psi$ be a vector in the candidate domain of the Stone generator (the submodule of vectors for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$). For any complex scalar $c \in \mathbb{C}$, the derivative of the orbit of the scaled vector $c\psi$ at $t=0$ is equal to $c$ times the derivative of the orbit of $\psi$ at $t=0$. That is: $$\left. \frac{d}{dt} U(t)(c\psi) \right|_{t=0} = c \left( \left. \frac{d}{dt} U(t)\psi \right|_{t=0} \right)$$

Given a one-parameter unitary group $U(t)$ on a Hilbert space $H$, let $D$ be the $\mathbb{C}$-submodule of vectors $\psi \in H$ for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$. This definition is the $\mathbb{C}$-linear map from $D$ to $H$ defined by $$\psi \mapsto -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$$ where $i$ is the imaginary unit.

Given a one-parameter unitary group $U(t)$ on a Hilbert space $H$, the candidate Stone generator is a partially defined $\mathbb{C}$-linear map from $H$ to $H$. Its domain is the $\mathbb{C}$-submodule of vectors $\psi \in H$ for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$. For any $\psi$ in this domain, the generator is defined as: $$\psi \mapsto -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$$ where $i$ is the imaginary unit.

Let $U(t)$ be a one-parameter unitary group on a Hilbert space $H$. For any vector $\psi$ in the domain of the candidate Stone generator (the $\mathbb{C}$-submodule of vectors for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t=0$), the value of the generator applied to $\psi$ is given by: $$H_{\text{gen}}\psi = -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$$ where $i$ is the imaginary unit and $\left. \frac{d}{dt} U(t)\psi \right|_{t=0}$ denotes the derivative of the orbit at $t=0$.

Let $\mathcal{H}$ be a complex Hilbert space and $U(t)$ be a one-parameter unitary group on $\mathcal{H}$. Let $A$ be the candidate Stone generator, a partially defined linear operator whose domain is the set of vectors $\psi \in \mathcal{H}$ for which the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$, with the action defined as $A\psi = -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$. Then $A$ is a symmetric operator, satisfying: $$\langle A\psi_1, \psi_2 \rangle = \langle \psi_1, A\psi_2 \rangle$$ for all $\psi_1, \psi_2$ in the domain of $A$, where $\langle \cdot, \cdot \rangle$ denotes the inner product on $\mathcal{H}$.