Physlib.QuantumMechanics.OneDimension.Operators.Unbounded

Let $\mathcal{H}$ be the one-dimensional Hilbert space. Given a topological vector space $S$ over $\mathbb{C}$ and an injective continuous linear map $\iota: S \to \mathcal{H}$ which embeds $S$ as a subobject (domain) of the Hilbert space, an unbounded operator is defined as a continuous linear map $op: S \to \mathcal{H}$.

This instance allows an unbounded operator $U$, defined on a domain $S$ which is embedded into a Hilbert space $\mathcal{H}$ via an injective continuous linear map $\iota: S \to \mathcal{H}$, to be treated as a function. Specifically, it enables the notation $U(s)$ for $s \in S$, mapping the element $s$ to its image in the Hilbert space $\mathcal{H}$.

Given a topological vector space $S$ over $\mathbb{C}$, a Hilbert space $\mathcal{H}$, and an injective continuous linear map $\iota : S \to \mathcal{H}$ which embeds $S$ as the domain of an operator, this definition constructs an unbounded operator from a continuous linear map $A: S \to S$. The resulting unbounded operator $T: S \to \mathcal{H}$ is defined by the composition $T = \iota \circ A$, mapping an element $\psi \in S$ to $\iota(A \psi) \in \mathcal{H}$.

Let $S$ be a topological vector space over $\mathbb{C}$ and $\mathcal{H}$ be a Hilbert space. Let $\iota: S \to \mathcal{H}$ be an injective continuous linear map that embeds $S$ into $\mathcal{H}$. For any continuous linear map $A: S \to S$, let $T$ be the corresponding unbounded operator from $S$ to $\mathcal{H}$ defined by the composition $T = \iota \circ A$. For any element $\psi \in S$, the value of the operator $T$ applied to $\psi$ is given by $T \psi = \iota(A \psi)$.

Let $S$ be a topological vector space over $\mathbb{C}$ and $\iota: S \to \mathcal{H}$ be a continuous injective embedding of $S$ into a Hilbert space $\mathcal{H}$. Given an unbounded operator $U: S \to \mathcal{H}$, a continuous linear functional $F: S \to \mathbb{C}$ is a **generalized eigenvector** of $U$ with eigenvalue $c \in \mathbb{C}$ if, for every $\psi \in S$, there exists an element $\psi' \in S$ such that $\iota(\psi') = U\psi$ and $F(\psi') = c F(\psi)$.

Let $S$ be a topological vector space over $\mathbb{C}$ and $\mathcal{H}$ be a Hilbert space, with $\iota: S \to \mathcal{H}$ being a continuous injective embedding. For any continuous linear map $A: S \to S$, let $U = \iota \circ A$ be the corresponding unbounded operator from $S$ to $\mathcal{H}$. A continuous linear functional $F: S \to \mathbb{C}$ is a generalized eigenvector of $U$ with eigenvalue $c \in \mathbb{C}$ if and only if for all $\psi \in S$, the condition $F(A \psi) = c F(\psi)$ holds.

Let $U: S \to \mathcal{H}$ be an unbounded operator with domain $S$ and embedding $\iota: S \to \mathcal{H}$. The operator $U$ is self-adjoint if for all $\psi_1, \psi_2 \in S$, the equality $$\langle U \psi_1, \iota \psi_2 \rangle_{\mathbb{C}} = \langle \iota \psi_1, U \psi_2 \rangle_{\mathbb{C}}$$ holds, where $\langle \cdot, \cdot \rangle_{\mathbb{C}}$ denotes the inner product on the Hilbert space $\mathcal{H}$.