Stone's theorem on one-parameter unitary groups

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 $H$ and one-parameter families

{(U_{t})}_{t \in R}

of unitary operators that are strongly continuous, i.e.,

\forall t_{0} \in R, ψ \in H : lim_{t \to t_{0}} U_{t} (ψ) = U_{t_{0}} (ψ)

If the self-adjoint operator is $A$ then the one-parameter subgroup is

U_{t} = e^{i t A}

See Wikipedia entry