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Concept
Stone's theorem on one-parameter unitary groups
Résumé
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families
of unitary operators that are strongly continuous, i.e.,
and are homomorphisms, i.e.,
Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.
The theorem was proved by , and showed that the requirement that be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.
This is an impressive result, as it allows one to define the derivative of the mapping which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.
The statement of the theorem is as follows.
Theorem. Let be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator , that is self-adjoint on and such that
The domain of is defined by
Conversely, let be a (possibly unbounded) self-adjoint operator on Then the one-parameter family of unitary operators defined by
is a strongly continuous one-parameter group.
In both parts of the theorem, the expression is defined by means of the spectral theorem for unbounded self-adjoint operators.
The operator is called the infinitesimal generator of Furthermore, will be a bounded operator if and only if the operator-valued mapping is norm-continuous.
The infinitesimal generator of a strongly continuous unitary group may be computed as
with the domain of consisting of those vectors for which the limit exists in the norm topology. That is to say, is equal to times the derivative of with respect to at . Part of the statement of the theorem is that this derivative exists—i.e., that is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since is only assumed (ahead of time) to be continuous, and not differentiable.
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