In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space ${\mathcal {H}}$ and one-parameter families
$$ \left(U_{t}\right)_{t \in \mathbb{R}} $$of unitary operators that are strongly continuous, i.e.,
$$ \forall t_{0} \in \mathbb{R}, \psi \in \mathcal{H}: \quad \lim _{t \rightarrow t_{0}} U_{t}(\psi)=U_{t_{0}}(\psi) $$If the self-adjoint operator is $A$ then the one-parameter subgroup is
$$ U_{t}=e^{i t A} $$See Wikipedia entry
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Author of the notes: Antonio J. Pan-Collantes
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