Stone's theorem on one-parameter unitary groups

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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

<math> \{U_t\}_{t \in \mathbb{R}} </math>

which are strongly continuous, that is

<math> \lim_{t \rightarrow t_0} U_t \xi = U_{t_0} \xi \quad \forall t_0 \in \mathbb{R}, \xi \in H </math>

and are homomorphisms:

<math> U_{t+s} = U_t U_s. \quad </math>

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.

The formal statement is as follows:

Theorem. Let A be a self-adjoint operator on a Hilbert space H. Then

<math> U_t = e^{i t A} \quad t \in \mathbb{R} </math>

is a strongly continuous one-parameter family of unitary operators. The infinitesimal generator of {U_t}_t is the operator i A. This mapping is a bijective correspondence.

A will be a bounded operator iff the operator-valued function t → U_t is norm continuous.

Example. The family of translation operators

<math> [T_t \psi](x) = \psi(x + t) \quad </math>

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

<math> \frac{d}{dx} = i \frac{1}{i} \frac{d}{dx} </math>

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

<math> T_t = e^{t \, {d}/{dx}}. \quad </math>

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 is a generalization of Stone's theorem to strongly continuous one-parameter semigroups of contractions on a 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)