Stone’s Theorem

Full treatment. Read Stone’s Theorem for the two proof directions, generator uniqueness, differentiable vectors, translations, and the nonautonomous boundary.

Statement

A family $U(t)$ is a strongly continuous one-parameter unitary group if and only if there is a unique self-adjoint $H$ such that

$$U(t)=e^{-itH/\hbar}.$$

The group determines the exact generator domain:

$$D(H) = \left\{ \psi: \lim_{t\to0} \frac{U(t)\psi-\psi}{t} \text{ exists in norm} \right\},$$

and

$$H\psi = i\hbar \lim_{t\to0} \frac{U(t)\psi-\psi}{t}.$$

Check before use

• Strong continuity means norm continuity of $U(t)\psi$ for each fixed vector.
• The generator is self-adjoint and may be unbounded.
• Every vector evolves continuously; only vectors in $D(H)$ need be strongly differentiable.
• Operator-norm continuity is equivalent to a bounded generator.
• Generic $H(t)$ produces a two-parameter propagator, not one autonomous group.

References

• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• M. H. Stone, “On one-parameter unitary groups in Hilbert space,” Annals of Mathematics 33, 643–648, 1932, doi:10.2307/1968538.