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# 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 which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators

$\left\{U_t\right\}_\left\{t in mathbb\left\{R\right\}\right\}$

which are strongly continuous, that is

$lim_\left\{t rightarrow t_0\right\} U_t xi = U_\left\{t_0\right\} xi quad forall t_0 in mathbb\left\{R\right\}, xi in H$

and are homomorphisms:

$U_\left\{t+s\right\} = U_t U_s. quad$

Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem is named after Marshall Stone who formulated and proved this theorem in 1932.

## Formal statement

Let U be a strongly continuous 1-parameter unitary group, then there exists a unique self-adjoint operator A such that

$U_t := e^\left\{i t A\right\} quad t in mathbb\left\{R\right\}.$

Conversely, let A be a self-adjoint operator on a Hilbert space H. Then

$U_t := e^\left\{i t A\right\} quad t in mathbb\left\{R\right\}$

is a strongly continuous one-parameter family of unitary operators.

The infinitesimal generator of {Ut}t is the operator $iA$. This mapping is a bijective correspondence. A will be a bounded operator iff the operator-valued function $t mapsto U_t$ is norm continuous.

## Example

The family of translation operators

$\left[T_t psi\right]\left(x\right) = psi\left(x + t\right) quad$

is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator

$frac\left\{d\right\}\left\{dx\right\} = i frac\left\{1\right\}\left\{i\right\} frac\left\{d\right\}\left\{dx\right\}$

defined on the space of complex-valued continuously differentiable functions of compact support on R. Thus

$T_t = e^\left\{t , \left\{d\right\}/\left\{dx\right\}\right\}. quad$

## Applications and generalizations

Stone's theorem has numerous applications in quantum mechanics. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on H. The infinitesimal generator of this group is the system Hamiltonian.

The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.

## References

• M. H. Stone, On one-parameter unitary groups in Hilbert Space, Annals of Mathematics 33, 643-648, (1932).
• K. Yosida, Functional Analysis, Springer-Verlag, (1968)
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