ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Position operator on L^2(R)

Example

Assume the Axiom of Choice. On $H = L^{2} (R)$ let $Q$ be the multiplication operator by the coordinate function $x$ , with domain $D (Q) = {f \in L^{2} (R) : \int_{R} x^{2} ∣ f (x) ∣^{2} d x < \infty}$ . Then $Q$ is self-adjoint with $σ (Q) = R$ , its spectral PVM is $E (B) f = 1_{B} f$ , and $U (t) f (x) = e^{i t x} f (x)$ defines a strongly continuous unitary group whose generator is $i Q$ ; in particular $D (Q)$ is a proper dense subspace of $H$ .

Facts & Assumptions

[A1]

On a $σ$ -finite measure space $(X, Σ, μ)$ , for a real measurable multiplier $m$ that is finite almost everywhere, the multiplication-operator example gives the domain, self-adjointness, spectral PVM $E (B) f = 1_{m^{- 1} (B)} f$ , functional calculus and essential-range spectrum formula on $L^{2} (X, μ)$ (Multiplication operators: domain, spectral measure and spectrum).

[A2]

A self-adjoint operator $T$ generates the strongly continuous unitary group $e^{i t T}$ computed by the Borel calculus, with generator $i T$ and derivative domain $D (T)$ (A self-adjoint operator generates a strongly continuous unitary group, Strongly continuous one-parameter unitary group).

Verification

technique · direct

Given: $H = L^{2} (R)$ and the multiplication operator $Q$ by $x$ .

1.1A1

$Q$ is the multiplication operator of the previous example for the measure space $(R, Borel, λ)$ and $m (x) = x$ : the domain, the self-adjointness, the spectral PVM $E (B) f = 1_{B} f$ and the calculus $g (Q) f = (g \circ m) f$ are those results.

1.2A1

The essential range of $x$ is $R$ , since every interval $(t - ε, t + ε)$ has positive Lebesgue measure, so $σ (Q) = R$ .

1.3A1A2

By the generation theorem applied to the self-adjoint operator $Q$ , the formula $U (t) = \int e^{i t x} d E (x)$ is a strongly continuous unitary group with generator $i Q$ , and the calculus of [A1] identifies $U (t) f (x) = e^{i t x} f (x)$ .

1.4A1

$D (Q)$ is proper and dense: it is dense by the previous example, and the function $f (x) = (1 + ∣ x ∣)^{- 1}$ for $∣ x ∣ \geq 1$ , extended by $1$ on $[- 1, 1]$ , lies in $L^{2} (R)$ but not in $D (Q)$ , because $\int_{1}^{\infty} x^{2} (1 + x)^{- 2} d x$ diverges.

2.1

The claims are steps 1.1, 1.2, 1.3 and 1.4. ∎

Depends on

Used by

A strongly continuous unitary group need not be norm continuous Counterexample

Dependency tree · two levels

43 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)