Physlib.Mathematics.OneParameterSubgroups.Unitary

For a complex Hilbert space $H$, this instance allows an element $U$ of the type `UnitaryOneParameterGroup H` to be treated as a function from the real numbers $\mathbb{R}$ to the space of bounded linear operators $\mathcal{B}(H)$. Specifically, it enables the notation $U(t)$ to represent the continuous linear map in the group corresponding to the parameter $t \in \mathbb{R}$.

Let $H$ be a complex Hilbert space. If $U$ and $V$ are unitary one-parameter groups on $H$ such that $U(t) = V(t)$ for all $t \in \mathbb{R}$, then $U = V$.

Let $U$ be a norm-continuous unitary one-parameter group on a complex Hilbert space $H$. For any $t \in \mathbb{R}$, the operator $U(t)$ is given by the exponential of its derivative at the origin: $$U(t) = \exp(t U'(0))$$ where $U'(0) = \left. \frac{d}{ds} U(s) \right|_{s=0}$ is the derivative of the map $t \mapsto U(t)$ evaluated at $t=0$.

Let $H$ be a complex Hilbert space and $U$ be a norm-continuous unitary one-parameter group on $H$. For any $t \in \mathbb{R}$, the adjoint of the operator $U(t)$, denoted $(U(t))^*$, is equal to the operator $U(-t)$.

For a norm-continuous unitary one-parameter group $U: \mathbb{R} \to \mathcal{B}(H)$ on a complex Hilbert space $H$, the generator $A$ is the bounded self-adjoint operator defined by $A = i \frac{d}{dt} U(t) \big|_{t=0}$. This definition corresponds to the physical convention where the group elements are expressed as $U(t) = e^{-itA}$.

Let $H$ be a complex Hilbert space and $U$ be a norm-continuous unitary one-parameter group on $H$. For any $t \in \mathbb{R}$, the operator $U(t)$ is given by the exponential of the operator $-itA$, where $A$ is the generator of $U$ and $i$ is the imaginary unit. That is, $U(t) = \exp(-itA)$.

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a norm-continuous unitary one-parameter group. If $A$ is a bounded linear operator on $H$ such that for all $t \in \mathbb{R}$, $U(t) = \exp(-itA)$, then $A$ is equal to the canonical generator of $U$. Here, $\exp$ denotes the operator exponential, $i$ is the imaginary unit, and the generator is defined as $i \left. \frac{d}{dt} U(t) \right|_{t=0}$.

For any unitary one-parameter group $U$ on a complex Hilbert space $H$, the adjoint of the derivative of $U$ evaluated at $t = 0$ is equal to the negative of that derivative. That is, $(U'(0))^* = -U'(0)$, where $U'(0) = \left. \frac{d}{dt} U(t) \right|_{t=0}$ and $^*$ denotes the adjoint operator.

For any norm-continuous unitary one-parameter group $U$ on a complex Hilbert space $H$, its infinitesimal generator $A$ is a self-adjoint operator.

Given a bounded self-adjoint operator $A$ on a complex Hilbert space $H$, this definition constructs the corresponding unitary one-parameter group $U: \mathbb{R} \to \mathcal{B}(H)$ defined by $U(t) = \exp(-itA)$, where $i$ is the imaginary unit and $\exp$ is the operator exponential.

Let $H$ be a complex Hilbert space and $A$ be a bounded self-adjoint operator on $H$. For any $t \in \mathbb{R}$, let $U(t)$ be the unitary one-parameter group generated by $A$ (denoted in the formal text as `ofSelfAdjoint hA`). Then the value of the group at $t$ is given by the operator exponential: $$U(t) = \exp(-itA)$$ where $i$ is the imaginary unit.

Let $H$ be a complex Hilbert space and $U$ be a norm-continuous unitary one-parameter group on $H$. Let $A$ be the infinitesimal generator of $U$ (the bounded self-adjoint operator such that $U$ is formally associated with $e^{-itA}$). Then the unitary one-parameter group constructed from $A$ via the operator exponential $t \mapsto \exp(-itA)$ is equal to $U$.

Let $H$ be a complex Hilbert space and $A$ be a bounded self-adjoint operator on $H$. If $U(t) = e^{-itA}$ is the unitary one-parameter group generated by $A$, then the generator of $U$ is equal to $A$.

This defines the equivalence (Stone's correspondence) between the type of norm-continuous unitary one-parameter groups $U: \mathbb{R} \to \mathcal{B}(H)$ on a complex Hilbert space $H$ and the type of bounded self-adjoint operators $A$ on $H$. Under this equivalence, a unitary one-parameter group corresponds to its infinitesimal generator $A$, and conversely, a self-adjoint operator $A$ corresponds to the group defined by $U(t) = e^{-itA}$.