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A self-adjoint operator generates a strongly continuous unitary group

Statement

Assume the Axiom of Choice. Let $T$ be a self-adjoint operator on $H$ with spectral projection valued measure $E$ , and set $U (t) : = e^{i t T} = \int_{R} e^{i t λ} d E (λ)$ for $t \in R$ . Then $U$ is a strongly continuous one-parameter unitary group, $U (t) D (T) = D (T)$ and $T U (t) = U (t) T$ on $D (T)$ , and the derivative $\lim_{t \to 0} \frac{1}{t} (U (t) x - x)$ exists exactly for $x \in D (T)$ , where it equals $i T x$ . Thus the generator of $U$ is $G = i T$ , with $D (G) = D (T)$ .

Facts & Assumptions

[A1]

For bounded Borel $h$ the operator $h (T)$ is bounded with $∥ h (T) x ∥^{2} = \int ∣ h ∣^{2} d E_{x}$ , $h (T)^{*} = \overline{h} (T)$ , and $h_{1} (T) h_{2} (T) = (h_{1} h_{2}) (T)$ ; the truncation definition gives $g (T) x = \lim_{m} g_{m} (T) x$ for $x \in D_{g}$ (Unbounded Borel functional calculus: domains, products, spectral mapping, The unbounded PVM integral is densely defined, closed and normal, Integral of a measurable function against a projection-valued measure).

[A2]

$D (T) = {x : \int λ^{2} d E_{x} < \infty}$ , and for $x \in D (T)$ one has $∥ (T - a) x ∥^{2} = \int (λ - a)^{2} d E_{x}$ for real $a$ ; also $U (t) D (T) = D (T)$ and $T U (t) = U (t) T$ on $D (T)$ , since $U (t) = e^{i t λ} (T)$ commutes with $E (B)$ and products of functions multiply (Spectral theorem for unbounded self-adjoint operators (PVM form), Unbounded Borel functional calculus: domains, products, spectral mapping).

[A3]

Scalar dominated convergence and Fatou's lemma apply to the finite measures $E_{x}$ (Dominated convergence, Fatou's lemma).

[A4]

The generator is defined by the difference quotients of the statement, and $D (G)$ is a linear subspace (Infinitesimal generator of a unitary group).

Proof

technique · direct

Given: A self-adjoint $T$ with spectral PVM $E$ , and $U (t) = e^{i t T}$ .

1.1

Unit and group law: $U (t)$ is bounded, $U (t) U (s) = (e^{i t λ} e^{i s λ}) (T) = U (t + s)$ and $U (0) = I$ by [A1], and $U (t)$ is unitary because $∥ U (t) x ∥^{2} = \int ∣ e^{i t λ} ∣^{2} d E_{x} = \int d E_{x} = ∥ x ∥^{2}$ and $U (t)^{*} = U (- t) = U (t)^{- 1}$ .

1.2

Strong continuity: for $x \in H$ and $t \to t_{0}$ , $∥ U (t) x - U (t_{0}) x ∥^{2} = \int ∣ e^{i t λ} - e^{i t_{0} λ} ∣^{2} d E_{x} \to 0$ by [A3], the integrand being bounded by $4$ and tending to $0$ pointwise.

A1A3

1.3

Derivative at $0$ for $x \in D (T)$ : $∥ \frac{1}{t} (U (t) x - x) - i T x ∥^{2} = \int ∣ \frac{1}{t} (e^{i t λ} - 1) - i λ ∣^{2} d E_{x} \to 0$ by [A3], since $∣ \frac{1}{t} (e^{i t λ} - 1) - i λ ∣ \leq 2 ∣ λ ∣$ and $∣ λ ∣^{2}$ is $E_{x}$ -integrable exactly because $x \in D (T)$ by [A2].

A2A3

2.1

Converse: if $z = \lim_{n} \frac{1}{t_{n}} (U (t_{n}) x - x)$ for some sequence $t_{n} \to 0$ , then by [A2] and Fatou's lemma $\int λ^{2} d E_{x} \leq {lim inf}_{n} \int ∣ \frac{1}{t_{n}} (e^{i t_{n} λ} - 1) ∣^{2} d E_{x} = {lim inf}_{n} ∥ \frac{1}{t_{n}} (U (t_{n}) x - x) ∥^{2} = ∥ z ∥^{2} < \infty$ , so $x \in D (T)$ ; step 1.3 then identifies the full limit as $i T x$ .

A2A3step 1.3

3.1

By steps 1.1, 1.2, 1.3 and 2.1 the family $U$ is a strongly continuous one-parameter unitary group whose generator satisfies $D (G) = D (T)$ and $G = i T$ ; the invariance and commutativity claims for $U (t)$ on $D (T)$ are [A2].

A2A4step 1.1step 1.2step 2.1∎

Depends on

Used by

Unitary groups converge under strong resolvent convergence Corollary
A strongly continuous unitary group need not be norm continuous Counterexample
Position operator on L²(R) Example
Stone's theorem: unitary groups and self-adjoint generators Theorem

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem (standard reference, not scraped)
Roland Schnaubelt, Evolution Equations (lecture notes) (standard reference, not scraped)