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

:$\{U\_t\}\_\{t\; in\; mathbb\{R$

which are strongly continuous, that is

:$lim\_\{t\; ightarrow\; t\_0\}\; U\_t\; xi\; =\; U\_\{t\_0\}\; xi\; quad\; forall\; t\_0\; in\; mathbb\{R\},\; xi\; in\; H$

and are homomorphisms:

:$U\_\{t+s\}\; =\; U\_t\; U\_s.\; quad$

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

:$U\_t\; :=\; e^\{i\; t\; A\}\; quad\; t\; in\; mathbb\{R\}.$

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

:$U\_t\; :=\; e^\{i\; t\; A\}\; quad\; t\; in\; mathbb\{R\}$

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

The infinitesimal generator of {"U"_"t"}_"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

:$[T\_t\; psi]\; (x)\; =\; psi(x\; +\; t)\; quad$

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

:$frac\{d\}\{dx\}\; =\; i\; frac\{1\}\{i\}\; frac\{d\}\{dx\}$

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

:$T\_t\; =\; e^\{t\; ,\; \{d\}/\{dx.\; quad$

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)