Stone's theorem on one-parameter unitary groups Guide, Meaning , Facts, Information and Description
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
which are
strongly continuous, that is
and are homomorphisms:
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
is a strongly continuous one-parameter family of unitary operators. The
infinitesimal generator of {
Ut}
t is the operator i
A. This mapping is a bijective correspondence.
A will be a bounded operator iff {Ut}t is norm continuous.
Example. The family of translation operators
is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator
defined on the space of complex-valued continuously differentiable functions of compact support on
R. Thus
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 infinitesiaml 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)
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