Manuscript Em03-F@1 · Em03-F-v1.tex · lemma · 0d4d148b

Lemma: Bounded Weyl expectation continuity

Controlled state classes and profile perturbations · lines 1398–1432 · proof included or linked

Em03-F@1#lem:weylcontinuity, Em03-F@1#eq:weylcontinuity

For $ω \in S_{C}$ ,

∣ ω (W (u)) - ω (W (l)) ∣ \leq C ∥ u - l ∥_{U} + \frac{∣ σ ( u , l ) ∣}{2} .

Consequently, for either the sine or cosine effect,

∣ ω (E_{∙, u}) - ω (E_{∙, l}) ∣ \leq H_{C} (l) ∥ u - l ∥_{U}, H_{C} (l) = \frac{C + ∥ l ∥ _{U}}{2} .

Proof. Use the Weyl relation to write $W (u)$ as a phase times $W (l) W (u - l)$ . Cauchy–Schwarz for the state and spectral calculus give

∣ ω (W (l) (W (u - l) - I)) ∣ \leq ω (X (u - l)^{2}) .

The phase costs at most $∣ σ (u, l) ∣/2$ . Finally,

∣ σ (u, l) ∣ = ∣ σ (u - l, l) ∣ \leq 2∥ u - l ∥_{U} ∥ l ∥_{U} .

Taking either real or imaginary parts and dividing by two proves the effect bound.

Displayed formulas in this passage

Em03-F@1#eq:weylcontinuity · line 1407
$∣ ω (E_{∙, u}) - ω (E_{∙, l}) ∣ \leq H_{C} (l) ∥ u - l ∥_{U}, H_{C} (l) = \frac{C + ∥ l ∥ _{U}}{2} .$
unlabeled · line 1401
$∣ ω (W (u)) - ω (W (l)) ∣ \leq C ∥ u - l ∥_{U} + \frac{∣ σ ( u , l ) ∣}{2} .$
unlabeled · line 1419
$∣ ω (W (l) (W (u - l) - I)) ∣ \leq ω (X (u - l)^{2}) .$
unlabeled · line 1425
$∣ σ (u, l) ∣ = ∣ σ (u - l, l) ∣ \leq 2∥ u - l ∥_{U} ∥ l ∥_{U} .$